Quadratic Equations

What is a quadratic equation?

A quadratic equation is an equation in which the highest power of the variable is 2. The standard form is:

ax² + bx + c = 0

Here a, b and c are numbers (the coefficients), and a must not be zero — otherwise the x² term disappears and you are left with a plain linear equation. If you are still warming up with equations where the variable is only to the first power, review solving linear equations first, because every method below relies on those balancing skills.

Some examples of quadratic equations:

• x² − 5x + 6 = 0 (here a = 1, b = −5, c = 6)
• 2x² + 3x − 2 = 0 (here a = 2, b = 3, c = −2)
• x² − 9 = 0 (here a = 1, b = 0, c = −9)

The word "quadratic" comes from the Latin quadratus, meaning square, because the variable is squared. When you draw a quadratic, you get a smooth U-shaped (or upside-down U) curve called a parabola.

Why two solutions?

Solving ax² + bx + c = 0 means finding the x-values that make the left side equal zero. Those points are exactly where the parabola crosses the x-axis. A U-shaped curve can cross the x-axis in two places, one place (just touching), or zero places (floating above it). That is why a quadratic usually has two solutions, sometimes one, and sometimes none (among real numbers). These solutions are also called the roots of the equation.

Method 1: Solving by square roots

When there is no bx term (b = 0), you can isolate x² and take the square root of both sides. If square roots are unfamiliar, the lesson on square numbers and square roots covers the basics.

Worked example: Solve x² − 9 = 0.

1. Add 9 to both sides: x² = 9.
2. Take the square root of both sides: x = ±√9.
3. Remember the plus-or-minus sign, because both 3² = 9 and (−3)² = 9.
4. So x = 3 or x = −3.

The ± is essential. Forgetting it is the most common mistake here, because it throws away one perfectly good answer.

Worked example: Solve 2x² − 50 = 0.

1. Add 50: 2x² = 50.
2. Divide by 2: x² = 25.
3. Square root both sides: x = ±√25 = ±5.
4. So x = 5 or x = −5.

Method 2: Solving by factoring

Factoring uses the zero-product rule: if two things multiply to give zero, then at least one of them must be zero. So if (x + 2)(x + 3) = 0, then either x + 2 = 0 or x + 3 = 0.

The trick is rewriting ax² + bx + c as a product of two brackets. When a = 1, look for two numbers that multiply to c and add to b.

Worked example: Solve x² − 5x + 6 = 0.

1. We need two numbers that multiply to +6 and add to −5.
2. Try the pairs: −2 and −3 give (−2)(−3) = 6 and (−2) + (−3) = −5.
3. So x² − 5x + 6 = (x − 2)(x − 3).
4. Set the equation to zero: (x − 2)(x − 3) = 0.
5. Apply the zero-product rule: x − 2 = 0 gives x = 2; x − 3 = 0 gives x = 3.
6. Solutions: x = 2 or x = 3.

Check: Substitute x = 2: (2)² − 5(2) + 6 = 4 − 10 + 6 = 0. Correct.

Worked example: Solve x² + 2x − 8 = 0.

1. Two numbers that multiply to −8 and add to +2: that is +4 and −2 (since 4 × −2 = −8 and 4 + −2 = 2).
2. Factor: (x + 4)(x − 2) = 0.
3. So x = −4 or x = 2.

Method 3: The quadratic formula

When factoring is awkward or the roots are not whole numbers, the quadratic formula always works. For any equation ax² + bx + c = 0:

x = ( −b ± √(b² − 4ac) ) / (2a)

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

Worked example: Solve 2x² + 3x − 2 = 0 using the formula.

1. Identify the coefficients: a = 2, b = 3, c = −2.
2. Compute the discriminant: b² − 4ac = 3² − 4(2)(−2) = 9 + 16 = 25.
3. Square root it: √25 = 5.
4. Substitute into the formula: x = (−3 ± 5) / (2 × 2) = (−3 ± 5) / 4.
5. The plus case: x = (−3 + 5)/4 = 2/4 = 0.5.
6. The minus case: x = (−3 − 5)/4 = −8/4 = −2.
7. Solutions: x = 0.5 or x = −2.

Worked example: Solve x² − 4x + 1 = 0 (this one does not factor nicely).

1. a = 1, b = −4, c = 1.
2. Discriminant: (−4)² − 4(1)(1) = 16 − 4 = 12.
3. √12 = 2√3 ≈ 3.46.
4. x = (4 ± 3.46) / 2.
5. Plus: x ≈ 7.46/2 ≈ 3.73. Minus: x ≈ 0.54/2 ≈ 0.27.
6. Solutions: x ≈ 3.73 or x ≈ 0.27.

Method 4: Completing the square

This method rewrites the equation so one side is a perfect square. It is also how the quadratic formula is derived.

Worked example: Solve x² + 6x + 5 = 0 by completing the square.

1. Move the constant: x² + 6x = −5.
2. Take half of the x-coefficient (half of 6 is 3) and square it (3² = 9).
3. Add 9 to both sides: x² + 6x + 9 = −5 + 9, so x² + 6x + 9 = 4.
4. The left side is now a perfect square: (x + 3)² = 4.
5. Square root both sides: x + 3 = ±2.
6. Solve: x = −3 + 2 = −1 or x = −3 − 2 = −5.
7. Solutions: x = −1 or x = −5.

The discriminant: a quick preview of the answer

The part under the square root, b² − 4ac, is called the discriminant. Before solving fully, it tells you what kind of answers to expect.

Discriminant (b² − 4ac)	Number of real solutions	What the graph does
Positive (> 0)	Two different real roots	Parabola crosses x-axis twice
Zero (= 0)	One repeated real root	Parabola just touches x-axis
Negative (< 0)	No real roots	Parabola never reaches x-axis

Example: For x² + 2x + 5 = 0, the discriminant is 2² − 4(1)(5) = 4 − 20 = −16. Negative, so there are no real solutions — the parabola floats above the x-axis.

Where quadratics are used

Quadratics describe anything that accelerates or curves. The path of a thrown ball, a kicked football, or a jet of water all follow parabolas. Engineers use quadratics to design bridges and satellite dishes; physicists use them to predict how far a falling object travels; and businesses use them to find the price that maximizes profit. Solving ax² + bx + c = 0 answers practical questions like "when does the ball hit the ground?" To see how these curves are drawn, head to functions and graphs.

Practice activity

Solve each equation. Use whichever method fits best.

1. x² − 16 = 0
2. x² + 7x + 12 = 0
3. x² − 3x − 10 = 0
4. 3x² − 12 = 0
5. x² − 2x + 1 = 0 (what do you notice?)

Answers:

1. x² = 16, so x = ±4. x = 4 or x = −4.
2. Two numbers multiplying to 12, adding to 7: 3 and 4. (x + 3)(x + 4) = 0, so x = −3 or x = −4.
3. Two numbers multiplying to −10, adding to −3: −5 and 2. (x − 5)(x + 2) = 0, so x = 5 or x = −2.
4. 3x² = 12, x² = 4, x = ±2.
5. (x − 1)² = 0, so x = 1 — a single repeated root (the discriminant is zero).

Summary

A quadratic equation has the form ax² + bx + c = 0. You can solve it by square roots (when b = 0), factoring (when factors are easy to spot), completing the square, or the all-purpose quadratic formula x = (−b ± √(b² − 4ac)) / (2a). The discriminant b² − 4ac tells you in advance whether to expect two, one, or no real solutions. Always check your answers by substituting them back into the original equation.