Work out the solutions to 7 + 2(5x + 2x^2) = 16, giving each answer as a surd in its simplest form.

Step-by-step solution

1. 1

   Expand the bracket and gather everything on one side.

   $7+10x+4x^{2}=16\quad\Longrightarrow\quad 4x^{2}+10x-9=0.$

   Write the terms in descending powers so the coefficients $a=4$, $b=10$, $c=-9$ are easy to read off.

2. 2

   Test for factorisation first. You would need two integers with product $4\cdot(-9)=-36$ and sum $10$; the closest pairs are $(-2,18)$ and $(12,-3)$, summing to $16$ and $9$. Nothing works, so the roots are irrational and the answer really will be a surd.

3. 3

   Compute the discriminant.

   $b^{2}-4ac=10^{2}-4(4)(-9)=100+144=244.$

4. 4

   Simplify the surd before dividing. $244=4\cdot 61$ and $61$ is prime, so

   $\sqrt{244}=2\sqrt{61}.$

   Simplifying here is what makes the final fraction reduce; leaving $\sqrt{244}$ in place gives a correct but unsimplified answer.

5. 5

   Apply the quadratic formula and cancel the common factor 2.

   $x=\frac{-10\pm 2\sqrt{61}}{2\cdot 4}=\frac{2\left(-5\pm\sqrt{61}\right)}{8}=\frac{-5\pm\sqrt{61}}{4}.$

6. 6

   Check with Vieta and a decimal estimate. The two roots sum to $\tfrac{-10}{4}=-2.5$ and multiply to $\tfrac{-9}{4}=-2.25$. Numerically $\sqrt{61}=7.8102$, so the roots are $0.70256$ and $-3.20256$; their sum is $-2.5$ and their product is $-2.25$, both correct.