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) $$ f(x) imes g(x)(2 $$ Marks $$ ) $$ (d) Solve for $$ x $$ in the equation $$ f(x)=g(x) $$. (3 Marks)

Question

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) $$ f(x) 	imes g(x)(2 $$ Marks $$ ) $$ (d) Solve for $$ x $$ in the equation $$ f(x)=g(x) $$. (3 Marks)

UpStudy AI · Accepted Answer

**(a)** We are given  
$$
f(x)=4x^2-7x+3 \quad \text{and} \quad g(x)=x-2.
$$  
Then,
$$
f(x)+g(x) = \left(4x^2-7x+3\right) + \left(x-2\right).
$$  
Combine like terms:  
$$
4x^2-7x+x+3-2 = 4x^2-6x+1.
$$

---

**(b)** Compute the difference:
$$
f(x)-g(x) = \left(4x^2-7x+3\right) - \left(x-2\right).
$$  
Distribute the negative sign and combine like terms:
$$
4x^2-7x+3 - x+2 = 4x^2-8x+5.
$$

---

**(c)** Multiply the functions:
$$
f(x)\times g(x) = \left(4x^2-7x+3\right)(x-2).
$$  
Multiply term-by-term:
- $$4x^2 \times x = 4x^3$$
- $$4x^2 \times (-2) = -8x^2$$
- $$-7x \times x = -7x^2$$
- $$-7x \times (-2) = 14x$$
- $$3 \times x = 3x$$
- $$3 \times (-2) = -6$$

Now, combine like terms:
$$
4x^3 + (-8x^2-7x^2) + (14x+3x) -6 = 4x^3 -15x^2 +17x -6.
$$

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**(d)** Solve for $$ x $$ in the equation $$ f(x)=g(x) $$:
$$
4x^2-7x+3 = x-2.
$$  
Bring all terms to one side:
$$
4x^2-7x+3 - x+2 = 4x^2-8x+5=0.
$$  
This is a quadratic equation of the form  
$$
ax^2+bx+c=0 \quad \text{with} \quad a=4,\ b=-8,\ c=5.
$$

Compute the discriminant:
$$
\Delta = b^2-4ac = (-8)^2-4\cdot4\cdot5 = 64-80 = -16.
$$

Since the discriminant is negative, the solutions are complex:
$$
x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{8 \pm \sqrt{-16}}{8} = \frac{8 \pm 4i}{8} = 1 \pm \frac{i}{2}.
$$

Thus, the solutions are:
$$
x=1+\frac{i}{2} \quad \text{and} \quad x=1-\frac{i}{2}.
$$
**(a)** $$ f(x) + g(x) = 4x^2 - 6x + 1 $$ **(b)** $$ f(x) - g(x) = 4x^2 - 8x + 5 $$ **(c)** $$ f(x) \times g(x) = 4x^3 - 15x^2 + 17x - 6 $$ **(d)** The solutions for $$ f(x) = g(x) $$ are $$ x = 1 + \frac{i}{2} $$ and $$ x = 1 - \frac{i}{2} $$.

If and , find in simplest form:
(a) Mark)
(b) Marks
© Marks
(d) Solve for in the equation . (3 Marks)

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If and , find in simplest form: (a) Mark) (b) Marks © Marks (d) Solve for in the equation . (3 Marks)