If \( f(x)=4 x^{\wedge} 2-7 x+3 \) and \( g(x)=x-2 \), find in simplest form: (a) \( f(x)+g(x)(1 \) Mark) (b) \( f(x)-g(x)(2 \) Marks \( ) \) (c) \( f(x) \times g(x)(2 \) Marks \( ) \) (d) Solve for \( x \) in the equation \( f(x)=g(x) \). (3 Marks)

To solve the problem, let’s break it down step by step. Given: \( f(x) = 4x^2 - 7x + 3 \) \( g(x) = x - 2 \) ### (a) \( f(x) + g(x) \) To add the functions, we substitute them into the expression: \[ f(x) + g(x) = (4x^2 - 7x + 3) + (x - 2) \] Combining the terms: \[ = 4x^2 - 7x + x + 3 - 2 \] \[ = 4x^2 - 6x + 1 \] So, the answer is: \( \boxed{4x^2 - 6x + 1} \) ### (b) \( f(x) - g(x) \) Now for the subtraction: \[ f(x) - g(x) = (4x^2 - 7x + 3) - (x - 2) \] Again combining the terms: \[ = 4x^2 - 7x + 3 - x + 2 \] \[ = 4x^2 - 8x + 5 \] So, the answer is: \( \boxed{4x^2 - 8x + 5} \) ### (c) \( f(x) \times g(x) \) Now, let’s multiply the two functions: \[ f(x) \times g(x) = (4x^2 - 7x + 3)(x - 2) \] Using the distributive property (also known as the FOIL method for binomials): \[ = 4x^2 \cdot x + 4x^2 \cdot (-2) - 7x \cdot x - 7x \cdot (-2) + 3 \cdot x + 3 \cdot (-2) \] \[ = 4x^3 - 8x^2 - 7x^2 + 14x + 3x - 6 \] \[ = 4x^3 - 15x^2 + 17x - 6 \] So, the answer is: \( \boxed{4x^3 - 15x^2 + 17x - 6} \) ### (d) Solve for \( x \) in the equation \( f(x) = g(x) \) This means we need to set \( f(x) \) equal to \( g(x) \): \[ 4x^2 - 7x + 3 = x - 2 \] Rearranging gives: \[ 4x^2 - 7x - x + 3 + 2 = 0 \] \[ 4x^2 - 8x + 5 = 0 \] Now we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 4, b = -8, c = 5 \): Calculating the discriminant: \[ b^2 - 4ac = (-8)^2 - 4 \cdot 4 \cdot 5 = 64 - 80 = -16 \] Since the discriminant is negative, there are no real solutions. Therefore, the solutions are complex: \[ x = \frac{8 \pm \sqrt{-16}}{8} = \frac{8 \pm 4i}{8} = 1 \pm \frac{1}{2}i \] So, the answer is: \( x = 1 + \frac{1}{2}i \) or \( x = 1 - \frac{1}{2}i \) Thus, the final answers are: (a) \( \boxed{4x^2 - 6x + 1} \) (b) \( \boxed{4x^2 - 8x + 5} \) (c) \( \boxed{4x^3 - 15x^2 + 17x - 6} \) (d) \( \boxed{1 + \frac{1}{2}i}, \, \boxed{1 - \frac{1}{2}i} \)