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Solving Quadratic Equations

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Solving Quadratic Equations

Solve each equation and show enough working to make your method clear. Give exact answers unless a question asks for a decimal. In this worksheet, roots are the values of x that make the equation equal to zero.

Set up the equation

Get each equation into standard form, ax² + bx + c = 0, before identifying its coefficients.

1.Rewrite 3x² + 5 = 8x in standard form. Then state a, b and c.

2.Which equation is quadratic?

4x − 9 = 0
x³ + 2x = 0
2x² − 7x + 1 = 0
5/x + 1 = 0

Factor and solve

Use factor pairs or a difference of squares. Remember: if a product is zero, at least one factor must be zero.

3.A puzzle gives the equation x² − 5x + 6 = 0. Factor it and find both possible values of x.

4.Solve x² − 49 = 0. Show how you obtain both roots.

5.A student factors x² + x − 12 = 0 as (x + 4)(x − 3) = 0 and writes x = −4 or x = 3. Is the solution correct? Explain by checking the factorisation.

Formula and discriminant

When a quadratic does not factor neatly, the quadratic formula works. The discriminant, b² − 4ac, tells you how many real roots to expect.

6.Use the quadratic formula to solve 2x² + 3x − 1 = 0. Give exact answers in surd form.

7.For x² − 4x + 7 = 0, calculate the discriminant and state the number of real solutions.

8.Match each equation to the number of distinct real roots: (i) x² − 6x + 9 = 0; (ii) x² − 5x + 6 = 0; (iii) x² + 2x + 5 = 0. Choose from: no real roots, one real root, two real roots.

(i) x² − 6x + 9 = 0
(ii) x² − 5x + 6 = 0
(iii) x² + 2x + 5 = 0

Two real roots
No real roots
One real root

9.Solve x² − 5x + 6 = 0. Then explain what the solutions represent in the equation.

10.For x² − 16 = 0, choose the most efficient method and solve the equation.

11.A ball's height above the ground is modelled by h = −5t² + 20t + 25, where h is in metres and t is in seconds. Solve −5t² + 20t + 25 = 0 and state when the ball hits the ground. Ignore any negative time.

12.A monic quadratic has roots x = −2 and x = 5. Write its factorised form and expand it into standard form.