Core Pure -as Year 1- Unit Test 5 Algebra And Functions

The quadratic equation $x^2 - 5x + 9 = 0$ has roots $\alpha$ and $\beta$. Find the value of: (a) $\alpha^2 + \beta^2$ (b) $\alpha^2\beta + \alpha\beta^2$

For reciprocal roots, simply reverse the coefficients: $2, -3, 4, -1$ becomes $-1, 4, -3, 2$ (watch the signs). core pure -as year 1- unit test 5 algebra and functions

Let $y = \frac1x$. Therefore $x = \frac1y$. Step 2: Substitute into the original equation: $$2\left(\frac1y\right)^3 - 3\left(\frac1y\right)^2 + 4\left(\frac1y\right) - 1 = 0$$ Step 3: Multiply through by $y^3$ to clear denominators: $$2 - 3y + 4y^2 - y^3 = 0$$ Step 4: Rearrange into standard polynomial form (highest power first): $$-y^3 + 4y^2 - 3y + 2 = 0$$ Or multiply by -1: $$y^3 - 4y^2 + 3y - 2 = 0$$ The quadratic equation $x^2 - 5x + 9

was a curveball—a partial fractions problem disguised as a rational function. Therefore $x = \frac1y$