Quadratic Equations: Question 4

Worked solution

Part (a): Factorise and solve $3x^2 + 11x - 20 = 0$

To factorise, find two numbers that multiply to $a\times c = 3\times(-20) = -60$ and add to $b = 11$.

Those numbers are $+15$ and $-4$, since $15\times(-4) = -60$ and $15 + (-4) = 11$.

Split the middle term:

$3x^2 + 15x - 4x - 20.$

Factorise in pairs:

$3x(x + 5) - 4(x + 5) = (3x - 4)(x + 5).$

So $3x^2 + 11x - 20 = (3x - 4)(x + 5)$.

Now set each factor equal to zero:

$3x - 4 = 0 \quad\Rightarrow\quad x = \frac{4}{3},$ $x + 5 = 0 \quad\Rightarrow\quad x = -5.$

Answer: $(3x - 4)(x + 5)$, giving $x = \dfrac{4}{3}$ or $x = -5$.

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Part (b): Solve $x^2 + 8x - 3 = 0$ by completing the square

The coefficient of $x^2$ is $1$, so take half of the coefficient of $x$. Half of $8$ is $4$, and $4^2 = 16$:

$x^2 + 8x - 3 = (x + 4)^2 - 16 - 3 = (x + 4)^2 - 19.$

So the equation becomes:

$(x + 4)^2 - 19 = 0,$

$(x + 4)^2 = 19.$

Take the square root of both sides, keeping both signs:

$x + 4 = \pm\sqrt{19},$

$x = -4 \pm \sqrt{19}.$

Now evaluate, using $\sqrt{19} = 4.3588\ldots$:

$x = -4 + 4.3588\ldots = 0.3588\ldots \approx 0.359,$ $x = -4 - 4.3588\ldots = -8.3588\ldots \approx -8.36.$

Answer (to 3 significant figures): $x = 0.359$ or $x = -8.36$.

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Part (c): Solve $2x^2 - 5x - 6 = 0$ using the quadratic formula

Here $a = 2$, $b = -5$ and $c = -6$. The quadratic formula is:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.$

First find the discriminant, taking care with the signs:

$b^2 - 4ac = (-5)^2 - 4(2)(-6) = 25 + 48 = 73.$

Substitute into the formula:

$x = \frac{-(-5) \pm \sqrt{73}}{2(2)} = \frac{5 \pm \sqrt{73}}{4}.$

Since $\sqrt{73} = 8.5440\ldots$:

$x = \frac{5 + 8.5440\ldots}{4} = \frac{13.5440\ldots}{4} = 3.3860\ldots \approx 3.39,$ $x = \frac{5 - 8.5440\ldots}{4} = \frac{-3.5440\ldots}{4} = -0.8860\ldots \approx -0.89.$

Answer (to 2 decimal places): $x = 3.39$ or $x = -0.89$.