Show that the equation $x^4 - x^2 - 9 = 0$ has a solution between $x = 1$ and $x = 2$ - OCR - GCSE Maths - Question 20 - 2019 - Paper 6

To find the solution, we will use the method of bisection.

Let’s evaluate the midpoint: $x = 1.5$ Calculating $f(1.5)$: $f(1.5) = (1.5)^4 - (1.5)^2 - 9 = 5.0625 - 2.25 - 9 = -6.1875$

As $f(1.5) < 0$, the root lies in $(1.5, 2)$.

Next, evaluate the midpoint of $(1.5, 2)$, which is: $x = 1.75$ Calculating $f(1.75)$: $f(1.75) = (1.75)^4 - (1.75)^2 - 9 = 9.3789 - 3.0625 - 9 = -2.6836$

Since $f(1.75) < 0$, the root lies in $(1.75, 2)$.

Now evaluate: $x = 1.875$ Calculating $f(1.875)$: $f(1.875) = (1.875)^4 - (1.875)^2 - 9 = 11.4102 - 3.5156 - 9 = -1.1054$

Continuing in this manner, we evaluate: $x = 1.9375$ Calculating $f(1.9375)$: $f(1.9375) = (1.9375)^4 - (1.9375)^2 - 9 = 12.8664 - 3.7461 - 9 = 0.1203$

Now, since $f(1.875) < 0$ and $f(1.9375) > 0$, the solution lies in $(1.875, 1.9375)$.

Repeating this process leads us to narrow down to: $x = 1.9$ Calculating: $f(1.9) = 1.9^4 - 1.9^2 - 9 = 12.0349 - 3.61 - 9 = -0.5751$

And finally evaluating: $f(1.91) = 12.1182 - 3.6681 - 9 = -0.5499$

Concluding that the solution is approximately: $x = 1.9$