Given 4k + 3r2 = 10, make r2 the subject of the formula. Also, for functions f(x) = (2x−7)/3 and g(x) = x…

Question

Given 4k + 3r2 = 10, make r2 the subject of the formula. Also, for functions f(x) = (2x−7)/3 and g(x) = x − 2/3, find f(1/2) and solve x when f(x) = g(x). Lastly, solve x2 + x = 132.

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Question (d)

Start with the given equation.
$4k + 3r^2 = 10$
Subtract $4k$ from both sides to isolate the term with $r^2$ .
$3r^2 = 10 - 4k$
Divide both sides by 3 to make $r^2$ the subject.
$r^2 = \frac{10 - 4k}{3}$ $\boxed{r^2 = \frac{10 - 4k}{3}}$
Question 8 (a) (i)
Substitute $x = \frac{1}{2}$ into the function $f(x) = \frac{2x-7}{3}$ .
$f\left(\frac{1}{2}\right) = \frac{2\left(\frac{1}{2}\right) - 7}{3}$
Simplify the numerator.
$f\left(\frac{1}{2}\right) = \frac{1 - 7}{3}$ $f\left(\frac{1}{2}\right) = \frac{-6}{3}$
Perform the division.
$f\left(\frac{1}{2}\right) = -2$ $\boxed{-2}$
Question 8 (a) (ii)
Set $f(x)$ equal to $g(x)$ .
$\frac{2x-7}{3} = x - \frac{2}{3}$
Multiply the entire equation by 3 to eliminate the denominators.
$3 \cdot \left(\frac{2x-7}{3}\right) = 3 \cdot \left(x - \frac{2}{3}\right)$ $2x - 7 = 3x - 2$
Rearrange the equation to solve for $x$ .
Subtract $2x$ from both sides.
$-7 = 3x - 2x - 2$ $-7 = x - 2$
Add 2 to both sides.
$-7 + 2 = x$ $x = -5$ $\boxed{x = -5}$
Question 8 (b)
Rearrange the quadratic equation into standard form $ax^2 + bx + c = 0$ .
$x^2 + x = 132$ $x^2 + x - 132 = 0$
Factor the quadratic expression.
We need two numbers that multiply to -132 and add to 1. These numbers are 12 and -11.
$(x + 12)(x - 11) = 0$
Set each factor equal to zero and solve for $x$ .
$x + 12 = 0 \quad \text{or} \quad x - 11 = 0$ $x = -12 \quad \text{or} \quad x = 11$ $\boxed{x = -12 \text{ or } x = 11}$

Given 4k + 3r2 = 10, make r2 the subject of the formula. Also, for functions f(x) = (2x−7)/3 and g(x) = x − 2/3, find f(1/2) and solve x when f(x) = g(x). Lastly, solve x2 + x = 132.

Step-by-step solution

Start with the given equation.

Subtract $4k$ from both sides to isolate the term with $r^2$ .

Divide both sides by 3 to make $r^2$ the subject.

Substitute $x = \frac{1}{2}$ into the function $f(x) = \frac{2x-7}{3}$ .

Simplify the numerator.

Perform the division.

Set $f(x)$ equal to $g(x)$ .

Multiply the entire equation by 3 to eliminate the denominators.

Rearrange the equation to solve for $x$ .

Add 2 to both sides.

Rearrange the quadratic equation into standard form $ax^2 + bx + c = 0$ .

Factor the quadratic expression.

Set each factor equal to zero and solve for $x$ .

Still stuck on this one?

More Mathematics Questions

Given 4k + 3r2 = 10, make r2 the subject of the formula. Also, for functions f(x) = (2x−7)/3 and g(x) = x − 2/3, find f(1/2) and solve x when f(x) = g(x). Lastly, solve x2 + x = 132.

Step-by-step solution

Start with the given equation.

Subtract 4k4k4k from both sides to isolate the term with r2r^2r2.

Divide both sides by 3 to make r2r^2r2 the subject.

Substitute x=12x = \frac{1}{2}x=21​ into the function f(x)=2x−73f(x) = \frac{2x-7}{3}f(x)=32x−7​.

Simplify the numerator.

Perform the division.

Set f(x)f(x)f(x) equal to g(x)g(x)g(x).

Multiply the entire equation by 3 to eliminate the denominators.

Rearrange the equation to solve for xxx.

Add 2 to both sides.

Rearrange the quadratic equation into standard form ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0.

Factor the quadratic expression.

Set each factor equal to zero and solve for xxx.

Still stuck on this one?

More Mathematics Questions

Subtract $4k$ from both sides to isolate the term with $r^2$ .

Divide both sides by 3 to make $r^2$ the subject.

Substitute $x = \frac{1}{2}$ into the function $f(x) = \frac{2x-7}{3}$ .

Set $f(x)$ equal to $g(x)$ .

Rearrange the equation to solve for $x$ .

Rearrange the quadratic equation into standard form $ax^2 + bx + c = 0$ .

Set each factor equal to zero and solve for $x$ .