Quadratic Formula: How To Use It, With 10 Worked Examples

What Is The Quadratic Formula?

The quadratic formula is a rule that gives the solutions, also called roots or zeros, of any quadratic equation. A quadratic equation is any equation where the highest power of x is 2.

Here is the formula again, written out:

x = (-b ± √(b² – 4ac)) / (2a)

Read it as: "x equals negative b, plus or minus the square root of b squared minus four a c, all over two a."

The whole top part sits over 2a, not just the square root.

Standard Form And The Values Of A, b And c

Before you can use the formula, the equation must be in standard form:

ax² + bx + c = 0

Three rules matter here:

• The right side must be 0. If it is not, move everything to the left first.
• Keep the signs. In 2x² – 7x + 3 = 0, b is -7, not 7.
• a cannot be 0. If a were 0, the x² term would vanish and you would have a linear equation, not a quadratic.

If a term is missing, its value is 0. In x² – 9 = 0, b = 0. In x² + 4x = 0, c = 0.

What The ± Sign Means

The ± symbol means "plus or minus." It tells you to work out the formula twice: once adding the square root and once subtracting it.

That is why most quadratics have two answers: their U-shaped graph, a parabola, can cross the x-axis twice.

How To Use The Quadratic Formula In 5 Steps

Using the quadratic formula comes down to five steps: write the equation in standard form, name a, b and c, find the discriminant, plug into the formula, and simplify both answers.

Here is each step with the equation x² + 5x + 6 = 0.

1. Write it in standard form. It already is: x² + 5x + 6 = 0.
2. Name a, b and c. a = 1, b = 5, c = 6.
3. Find the discriminant, b² – 4ac. 5² – 4(1)(6) = 25 – 24 = 1.
4. Plug into the formula. x = (-5 ± √1) / (2 × 1) = (-5 ± 1) / 2
5. Split and simplify.

• With plus: (-5 + 1) / 2 = -4 / 2 = -2
• With minus: (-5 – 1) / 2 = -6 / 2 = -3

Check: put x = -2 back in: (-2)² + 5(-2) + 6 = 4 – 10 + 6 = 0. It works.

Finding the discriminant first tells you what kind of answer to expect.

The Discriminant: Know Your Answer Type First

The discriminant is the part under the square root sign, b² – 4ac. Its sign tells you how many real solutions the equation has before you solve it.

It is also a factoring test. If the discriminant is a perfect square (0, 1, 4, 9, 16 and so on), the equation factors neatly. If not, use the formula.

Quadratic Formula Examples

The 10 quadratic formula examples below cover every situation you will meet in class: whole-number answers, fractions, a repeated root, square roots, complex numbers, missing terms, rearranging, decimals, a negative a and two word problems. Each one shows every step.

Example 1: Two Whole-Number Answers

Solve x² – x – 12 = 0

• a = 1, b = -1, c = -12
• Discriminant: (-1)² – 4(1)(-12) = 1 + 48 = 49
• x = (1 ± √49) / 2 = (1 ± 7) / 2
• x = 8/2 = 4 or x = -6/2 = -3

Notice that -b is +1 here, because b is -1. The negative of a negative is positive.

Example 2: Fraction Answers

Solve 2x² – 7x + 3 = 0

• a = 2, b = -7, c = 3
• Discriminant: (-7)² – 4(2)(3) = 49 – 24 = 25
• x = (7 ± √25) / 4 = (7 ± 5) / 4
• x = 12/4 = 3 or x = 2/4 = 1/2

Always reduce fraction answers. 2/4 becomes 1/2, which a fraction simplifier can confirm in a second.

Example 3: One Repeated Answer

Solve x² – 6x + 9 = 0

• a = 1, b = -6, c = 9
• Discriminant: (-6)² – 4(1)(9) = 36 – 36 = 0
• x = (6 ± √0) / 2 = 6 / 2
• x = 3 (a double root)

When the discriminant is 0, plus and minus give the same answer. The parabola just touches the x-axis at x = 3. This equation is also a perfect square: (x – 3)² = 0.

Example 4: Answers With A Square Root

Solve x² + 4x – 1 = 0

• a = 1, b = 4, c = -1
• Discriminant: 4² – 4(1)(-1) = 16 + 4 = 20
• x = (-4 ± √20) / 2
• Simplify √20: √20 = √(4 × 5) = 2√5
• x = (-4 ± 2√5) / 2 = -2 ± √5
• As decimals: x ≈ 0.236 or x ≈ -4.236

To simplify, divide every term on top by 2: -4/2 = -2 and 2√5/2 = √5. A square root calculator helps when you need the decimal value of √5 (about 2.236).

Example 5: Complex Answers (Negative Discriminant)

Solve x² + 2x + 5 = 0

• a = 1, b = 2, c = 5
• Discriminant: 2² – 4(1)(5) = 4 – 20 = -16
• x = (-2 ± √(-16)) / 2
• √(-16) = 4i, where i is the imaginary unit (i² = -1)
• x = (-2 ± 4i) / 2 = -1 ± 2i

If your class has not covered complex numbers, the answer is "no real solutions."

Example 6: No x Term (b = 0)

Solve 3x² – 12 = 0

• a = 3, b = 0, c = -12
• Discriminant: 0² – 4(3)(-12) = 144
• x = (0 ± √144) / 6 = ±12 / 6
• x = 2 or x = -2

Faster: add 12, divide by 3 to get x² = 4, so x = ±2.

Example 7: No Constant Term (c = 0)

Solve 4x² + 8x = 0

• a = 4, b = 8, c = 0
• Discriminant: 8² – 4(4)(0) = 64
• x = (-8 ± 8) / 8
• x = 0/8 = 0 or x = -16/8 = -2

When c is 0, one answer is always 0.

Example 8: Rearrange First

Solve 2x² = 3x + 5

The right side is not 0, so subtract 3x and 5 from both sides:

• 2x² – 3x – 5 = 0
• a = 2, b = -3, c = -5
• Discriminant: (-3)² – 4(2)(-5) = 9 + 40 = 49
• x = (3 ± 7) / 4
• x = 10/4 = 5/2 or x = -4/4 = -1

Using a = 2, b = 3 and c = 5 straight from the original would give a wrong answer.

Example 9: Decimal Coefficients

Solve 0.5x² + 1.5x – 2 = 0

You can plug the decimals straight in, but it is easier to clear them first. Multiply every term by 2:

• x² + 3x – 4 = 0
• a = 1, b = 3, c = -4
• Discriminant: 3² – 4(1)(-4) = 9 + 16 = 25
• x = (-3 ± 5) / 2
• x = 1 or x = -4

Multiplying the whole equation by the same number does not change its solutions.

Example 10: A Negative Value Of A

Solve -x² + 4x + 5 = 0

• a = -1, b = 4, c = 5
• Discriminant: 4² – 4(-1)(5) = 16 + 20 = 36
• x = (-4 ± 6) / (2 × -1) = (-4 ± 6) / -2
• x = 2/-2 = -1 or x = -10/-2 = 5

Watch the denominator: 2a is -2. You can also multiply by -1 first to get x² – 4x – 5 = 0, which has the same solutions.

Word Problems Solved With The Quadratic Formula

Quadratic equations show up whenever something falls under gravity or covers an area.

A Ball Thrown Into The Air

A ball is thrown upward from 4 feet above the ground at 64 feet per second. Its height after t seconds is h = -16t² + 64t + 4. When does it hit the ground?

The ball hits the ground when h = 0:

• -16t² + 64t + 4 = 0
• a = -16, b = 64, c = 4
• Discriminant: 64² – 4(-16)(4) = 4,096 + 256 = 4,352
• √4,352 ≈ 65.97
• t = (-64 ± 65.97) / -32
• t ≈ -0.06 or t ≈ 4.06 seconds

Time cannot be negative, so the ball lands after about 4.06 seconds. The -16 is half of gravity's pull in feet (about 32 feet per second squared).

Finding The Size Of A Rectangle

A garden is 3 feet longer than it is wide. Its area is 40 square feet. How wide is it?

• Let w = width, so length = w + 3
• Area: w(w + 3) = 40
• Standard form: w² + 3w – 40 = 0
• a = 1, b = 3, c = -40
• Discriminant: 9 + 160 = 169
• w = (-3 ± 13) / 2
• w = 5 or w = -8

A width cannot be negative, so the garden is 5 feet wide and 8 feet long. Check: 5 × 8 = 40. ✓

How The Quadratic Formula Is Derived

The quadratic formula comes from solving ax² + bx + c = 0 by completing the square. Doing that once, with letters instead of numbers, gives a result that works for every quadratic.

1. Start with ax² + bx + c = 0.
2. Divide every term by a: x² + (b/a)x + c/a = 0
3. Move the constant: x² + (b/a)x = -c/a
4. Add (b/2a)² to both sides to complete the square: x² + (b/a)x + b²/4a² = b²/4a² – c/a
5. Write the left side as a square: (x + b/2a)² = (b² – 4ac) / 4a²
6. Take the square root of both sides: x + b/2a = ±√(b² – 4ac) / 2a
7. Subtract b/2a: x = (-b ± √(b² – 4ac)) / 2a

Quadratic Formula Vs Other Solving Methods

The quadratic formula always works, but it is not always the fastest. Pick the method that fits the equation in front of you.

If no factors jump out quickly, switch to the formula. A graphing calculator is handy for seeing where the parabola crosses the x-axis, so you can tell at a glance whether your answers make sense.

Two Fast Ways To Check Your Answers

Use substitution for certainty, or the sum and product test for speed.

Substitute Back In

Put each answer into the original equation. If the result is 0, it is correct.

For 2x² – 7x + 3 = 0 with x = 1/2: 2(1/4) – 7(1/2) + 3 = 0.5 – 3.5 + 3 = 0 ✓

Use The Sum And Product Of The Roots

For any quadratic ax² + bx + c = 0, the two solutions always follow these rules:

• Sum of the roots = -b/a
• Product of the roots = c/a

For 2x² – 7x + 3 = 0, the answers were 3 and 1/2:

• Sum: 3 + 1/2 = 7/2, and -b/a = 7/2 ✓
• Product: 3 × 1/2 = 3/2, and c/a = 3/2 ✓

It works with square roots too: in Example 4, the √5 parts cancel when you add, leaving -4 = -b/a.

How The Formula Connects To The Graph

The solutions from the quadratic formula are the x-intercepts of the parabola y = ax² + bx + c. The formula also hides the parabola's axis of symmetry.

Split the formula into -b/2a and ±√(b² – 4ac)/2a. The first piece is the axis of symmetry and the vertex's x-value. The second is how far each root sits from that line. For x² + 4x – 1 = 0, the axis is x = -2, and the roots sit √5 units on either side of it.

Common Mistakes To Avoid

Most quadratic formula errors are sign slips or dividing only part of the top by 2a. This table shows each mistake and how to avoid it.

Calculator tip: type negative numbers in parentheses and put the whole top in parentheses before dividing. Enter (-(-7) + √((-7)² – 4×2×3)) / (2×2), not -(-7) + √(…) / 2×2.

Practice Problems With Answers

Try these on your own, then check below. They are in order from easiest to hardest.

1. x² + 7x + 10 = 0
2. 3x² + 2x – 1 = 0
3. x² – 10x + 25 = 0
4. x² – 2x – 4 = 0
5. 2x² + 5x – 3 = 0
6. 5x² + 3x + 1 = 0

Answers

1. x = -2 or x = -5 (discriminant 9)
2. x = 1/3 or x = -1 (discriminant 16)
3. x = 5, a double root (discriminant 0)
4. x = 1 ± √5, about 3.236 and -1.236 (discriminant 20)
5. x = 1/2 or x = -3 (discriminant 49)
6. x = (-3 ± i√11) / 10, no real solutions (discriminant -11)

You can check any of these with the quadratic equation solver, which shows the discriminant and every step.

Summary

The quadratic formula, x = (-b ± √(b² – 4ac)) / 2a, solves every equation in the form ax² + bx + c = 0. Put the equation in standard form, name a, b and c with their signs, find the discriminant, then work out the plus and minus answers.

The discriminant tells you what to expect: two real roots if it is positive, one if it is zero, and complex roots if it is negative. Check answers by substituting them back in or with the sum and product rules.

Next step: work through the six practice problems above, then try writing your own word problem using area or a thrown ball. For more algebra help, the algebra calculator solves and explains other equation types.

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