Quadratic Equations

What is a quadratic equation?

A quadratic equation has the form ax^2 + bx + c = 0, where a, b and c are constants and a is not 0. Its solutions are called roots. Quadratics model areas, projectile paths and profit in business, so the KICD Senior School Mathematics course treats them in detail.

Method 1: Factorisation

Find two numbers whose product is ac and sum is b.

Example 1: Solve x^2 + 7x + 12 = 0.

• Numbers with product 12 and sum 7: 3 and 4.
• (x + 3)(x + 4) = 0, so x = -3 or x = -4.

Example 2: Solve 2x^2 + 5x - 3 = 0.

• ac = -6; numbers 6 and -1.
• 2x^2 + 6x - x - 3 = 0, 2x(x + 3) - 1(x + 3) = 0, (2x - 1)(x + 3) = 0.
• x = 1/2 or x = -3.

Special case, difference of two squares: x^2 - 25 = (x - 5)(x + 5), so x = 5 or -5.

Method 2: Completing the square

Solve x^2 + 6x - 7 = 0.

• x^2 + 6x = 7
• Add (6/2)^2 = 9: x^2 + 6x + 9 = 16
• (x + 3)^2 = 16, so x + 3 = 4 or -4
• x = 1 or x = -7

Method 3: Quadratic formula

x = (-b +/- sqrt(b^2 - 4ac)) / 2a

Example 3: Solve 3x^2 - 5x - 2 = 0 (a = 3, b = -5, c = -2).

• b^2 - 4ac = 25 + 24 = 49, square root = 7
• x = (5 + 7) / 6 = 2 or x = (5 - 7) / 6 = -1/3

The discriminant

D = b^2 - 4ac

• D > 0: two different real roots
• D = 0: one repeated root
• D < 0: no real roots (graph does not touch the x-axis)

Example 4: For x^2 - 6x + 9 = 0, D = 36 - 36 = 0, so x = 3 only.

Sum and product of roots

For ax^2 + bx + c = 0: sum = -b/a, product = c/a. Form an equation with roots 2 and 5: x^2 - (2 + 5)x + (2 x 5) = 0, so x^2 - 7x + 10 = 0.

Word problems

Example 5: The product of two consecutive positive integers is 56. Find them.

• x(x + 1) = 56, x^2 + x - 56 = 0
• (x + 8)(x - 7) = 0, x = 7 (reject -8)
• The numbers are 7 and 8.

Example 6 (Kenyan context): A tailor makes x shirts. Profit P = -x^2 + 20x - 36 (KES hundreds). Find x when profit is zero.

• x^2 - 20x + 36 = 0 gives (x - 18)(x - 2) = 0, so x = 2 or 18.
• Maximum profit is at x = 10: P = -100 + 200 - 36 = 64.

The graph

The graph of y = ax^2 + bx + c is a parabola. It opens upward if a > 0 and downward if a < 0. The axis of symmetry is x = -b/2a and the turning point is at that x-value.

Common errors

• Not rearranging to zero before factorising.
• Forgetting the negative sign in -b.
• Keeping answers that make no sense in context (such as negative lengths).

Try it

• Solve x^2 - 9x + 20 = 0.
• Solve 2x^2 - x - 6 = 0 using the formula.
• Find the value of k for which x^2 + kx + 16 = 0 has equal roots (take k positive).
• A ball's height is h = 20t - 5t^2. When does it hit the ground?