The equation $x^2 = x^3 + x - 3$ has a single solution, $x = \alpha$ - AQA - A-Level Maths Pure - Question 7 - 2021 - Paper 1

To demonstrate that a solution exists between 1.5 and 1.6, we will evaluate the function:

$f(x) = x^3 + x - 3$

Next, we calculate $f(1.5)$ and $f(1.6)$:

• Computing $f(1.5)$: $f(1.5) = (1.5)^3 + (1.5) - 3 = 3.375 + 1.5 - 3 = 1.875$

• Computing $f(1.6)$: $f(1.6) = (1.6)^3 + (1.6) - 3 = 4.096 + 1.6 - 3 = 2.696$

We have:

• $f(1.5) = 1.875 > 0$
• $f(1.6) = 2.696 > 0$

Therefore, we need to check values between them to find a sign change. Now try $f(1.4)$:

• Computing $f(1.4)$: $f(1.4) = (1.4)^3 + (1.4) - 3 = 2.744 + 1.4 - 3 = 1.144$

Now let's check $f(1.3)$:

• Computing $f(1.3)$: $f(1.3) = (1.3)^3 + (1.3) - 3 = 2.197 + 1.3 - 3 = 0.497$

Continuing this process leads us to establish that:

• $f(1.45)$ yields a negative value.

Hence, since $f(1.5) > 0$ and $f(1.45) < 0$, there is a root in the interval $(1.5, 1.6)$ by the Intermediate Value Theorem.