Quadratic Equations: Question 2

Worked solution

Part (a): Forming the quadratic

At the points of intersection $P$ and $Q$, the $y$-values of the curve and the line are equal:

$3x^2 - 2x = 5x + 5.$

Move every term to the left-hand side so that the right-hand side is $0$:

$3x^2 - 2x - 5x - 5 = 0,$

$\boxed{3x^2 - 7x - 5 = 0.} \quad \checkmark$

This is the required equation, with $a = 3$, $b = -7$ and $c = -5$.

Part (b): Solving with the quadratic formula

The quadratic formula is

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

Step 1: Evaluate the discriminant. Substituting $a = 3$, $b = -7$, $c = -5$:

$b^2 - 4ac = (-7)^2 - 4 \times 3 \times (-5) = 49 + 60 = 109.$

Since $109 > 0$ and is not a perfect square, there are two distinct irrational roots, consistent with the line cutting the curve at two separate points. Take the square root:

$\sqrt{109} = 10.4403\ldots$

Step 2: Substitute into the formula.

$x = \frac{-(-7) \pm \sqrt{109}}{2 \times 3} = \frac{7 \pm 10.4403\ldots}{6}.$

Step 3: Work out each root separately.

$x = \frac{7 + 10.4403\ldots}{6} = \frac{17.4403\ldots}{6} = 2.9067\ldots,$

$x = \frac{7 - 10.4403\ldots}{6} = \frac{-3.4403\ldots}{6} = -0.5733\ldots.$

Step 4: Round to 2 decimal places.

$x = 2.91 \qquad \text{or} \qquad x = -0.57.$

Check (alternative quick verification)

Substitute the more “surprising” negative root back into $3x^2 - 7x - 5$ using the un-rounded value $x = -0.5734$:

$3(-0.5734)^2 - 7(-0.5734) - 5 = 0.9864 + 4.0138 - 5 \approx 0.00. \checkmark$

(The tiny residual is only rounding error, confirming the root.) As a further sense-check, the two roots should sum to $-\dfrac{b}{a} = \dfrac{7}{3} \approx 2.333$, and indeed $2.9067 + (-0.5734) = 2.333\ldots$ ✓, while their product should be $\dfrac{c}{a} = -\dfrac{5}{3} \approx -1.667$, and $2.9067 \times (-0.5734) \approx -1.667$ ✓.

Final answer

$\boxed{x = 2.91 \quad \text{or} \quad x = -0.57 \ \text{(to 2 d.p.)}}$