Quadratic Formula & Discriminant (Ex 5J)

Year 10 Mathematics Core · Cambridge Ch 5, §5J · Mr Wong

Name

Class

Date

Learning intentions.

Know and apply the quadratic formula
Use the discriminant $\Delta=b^2-4ac$ to find the number of real solutions
Solve quadratic equations exactly (surd answers)

Key results

$x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$. Discriminant $\Delta=b^2-4ac$: $\Delta>0$ → two solutions, $\Delta=0$ → one solution, $\Delta<0$ → no real solutions.

Warm-up — find the discriminant

1. Find $\Delta=b^2-4ac$ and state the number of real solutions.

a $x^2-4x+4=0$

b $x^2+2x+5=0$

c $2x^2+3x-1=0$

d $x^2-6x+9=0$

Part A — Solving with the formula

2. Solve, leaving answers exact.

a $x^2+3x-3=0$

b $x^2-4x+1=0$

Part B — Non-monic with the formula

3. Solve, leaving answers exact.

a $2x^2+5x+1=0$

b $3x^2-2x-2=0$

5J — continued

Part C (number of solutions) · Part D (mixed) · Challenge

Name

Part C — How many solutions?

4. Without solving, use the discriminant to state the number of real solutions.

a $x^2+5x-3=0$

b $2x^2-3x+4=0$

c $x^2+6x+9=0$

d $x^2+7x-1=0$

Part D — Mixed

5. Solve $x^2+5x+3=0$ with the quadratic formula. Leave the answer exact.

6. Solve $2x^2-2x-1=0$ with the quadratic formula and simplify the surd fully.

Challenge

7. For what value of $k$ does $x^2+kx+9=0$ have exactly one real solution? (Set $\Delta=0$ and solve for $k$.)