The Quadratic Formula

Example 1: Using the Quadratic Formula

Find the solutions to the quadratic equation $3x^2 – 8x + 2=0$, giving your answers in surd form.

[2 marks]

$a = 3$, $b = -8$ and $c = 2$

Put these values into the quadratic formula:

$begin{aligned} x &= dfrac{-(-8) pm sqrt{(-8)^2 – 4 times 3 times 2}}{2 times 3} &= dfrac{8 pm sqrt{40}}{6} &= dfrac{4 pm sqrt{10}}{3} end{aligned}$

So, $x = textcolor{orange}{dfrac{4 + sqrt{10}}{3}}$ or $textcolor{orange}{x = dfrac{4 – sqrt{10}}{3}}$

Example 2: Finding the Discriminant

How many real roots does the quadratic equation $2x^2 – 5x + 3=0$ have?

[2 marks]

$a = 2$, $b = -5$ and $c = 3$

So, the discriminant is

$b^2-4ac = (-5)^2 – 4 times 2 times 3 = 1$

The discriminant is $>0$, so $2x^2 – 5x + 3=0$ has two real roots.

Example 3: Using the Discriminant

$f(x) = 2x^2 + 4x + k$. Find the values of $k$ for which $f(x)=0$ has no real roots.

[3 marks]

$a = 2$, $b = 4$ and $c = k$

The discriminant is

$b^2 – 4ac = 4^2 – 4 times 2 times k = 16 – 8k$

The quadratic equation has no real roots, therefore $b^2 – 4ac < 0$

So,

$begin{aligned} 16 – 8k &< 0 16 &< 8k textcolor{orange}{k} &textcolor{orange}{> 2} end{aligned}$

Example 4: Using the Discriminant

$kx^2 + 2kx + 3 = 0$ has two distinct real roots.

Find the set of values for which $k$ satisfies this.

[5 marks]

$a = k$, $b = 2k$ and $c = 3$

The discriminant is

$b^2 – 4ac = (2k)^2 – (4 times k times 3) = 4k^2 – 12k$

Since the quadratic equation has two distinct real roots, the discriminant must be $>0$

$4k^2 – 12k > 0$

The quadratic can be factorised:

$4k^2 – 12k = 4k(k-3)$

This is $0$ when $k=0$ or when $k=3$

Hence, $4k(k-3)>0$ when $textcolor{orange}{k<0}$ or when $textcolor{orange}{k>3}$

Note: You can draw a graph of $y = 4k(k-3)$ to help – you would see a u-shaped graph that crosses the $k$-axis at $0$ and $3$ (see inequalities).