QUESTION 3: Solve for a and b in the equations below: a + 3/b = 4 a − 12/b = −3

Question

ScanSolve AI Tutor · Accepted Answer

Here are the solutions to the questions.

QUESTION 1

a) Evaluate:

(i) $(3^2 \times 7^0)^3$

Evaluate the terms inside the parenthesis.
Recall that any non-zero number raised to the power of 0 is 1. So, $7^0 = 1$ .
$3^2 = 9$ $7^0 = 1$
Substitute these values back into the expression.
$(9 \times 1)^3$ $9^3$
Calculate the final value.
$9^3 = 9 \times 9 \times 9 = 81 \times 9 = 729$
The value is $\boxed{\text{729}}$ .

(ii) $(0.56)^0$
Apply the rule for exponents.
Any non-zero number raised to the power of 0 is 1.
$(0.56)^0 = 1$
The value is $\boxed{\text{1}}$ .

b) Find the value of $\sqrt{81}$
Find the number that, when multiplied by itself, equals 81.
$\sqrt{81} = 9$
The value is $\boxed{\text{9}}$ .

c) Express 0.00698 in standard form.
Move the decimal point to the right until the number is between 1 and 10.
The number becomes 6.98.
Count how many places the decimal point was moved.
The decimal point was moved 3 places to the right.
Write the number in standard form ( $A \times 10^n$ ).
Since the decimal was moved to the right, the exponent $n$ is negative.
$0.00698 = 6.98 \times 10^{-3}$
The standard form is $\boxed{6.98 \times 10^{-3}}$ .

QUESTION 2

a) Derive the quadratic formula by completing the square method, given an equation $ax^2 + bx + c = 0$ , where $a, b$ and $c$ are constants.
Start with the general quadratic equation and divide by $a$ (assuming $a \neq 0$ ).
$ax^2 + bx + c = 0$ $x^2 + \frac{b}{a}x + \frac{c}{a} = 0$
Move the constant term to the right side of the equation.
$x^2 + \frac{b}{a}x = -\frac{c}{a}$
Complete the square on the left side by adding $(\frac{b}{2a})^2$ to both sides.
$x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2$
Factor the left side as a perfect square and simplify the right side.
$\left(x + \frac{b}{2a}\right)^2 = -\frac{c}{a} + \frac{b^2}{4a^2}$
To combine the terms on the right, find a common denominator:
$\left(x + \frac{b}{2a}\right)^2 = -\frac{4ac}{4a^2} + \frac{b^2}{4a^2}$ $\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}$
Take the square root of both sides.
$x + \frac{b}{2a} = \pm\sqrt{\frac{b^2 - 4ac}{4a^2}}$ $x + \frac{b}{2a} = \pm\frac{\sqrt{b^2 - 4ac}}{\sqrt{4a^2}}$ $x + \frac{b}{2a} = \pm\frac{\sqrt{b^2 - 4ac}}{2a}$
Isolate $x$ to obtain the quadratic formula.
$x = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a}$ $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
The quadratic formula is $\boxed{x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}$ .

b) Solve for $a$ and $b$ in the equations below:
$\frac{9}{a} + \frac{12}{b} = -3 \quad (1)$ $\frac{3}{a} + \frac{6}{b} = 4 \quad (2)$
Let $x = \frac{1}{a}$ and $y = \frac{1}{b}$ to transform the system into linear equations.
$9x + 12y = -3 \quad (1')$ $3x + 6y = 4 \quad (2')$
Use the elimination method to solve for $x$ and $y$ .
Multiply equation (2') by 3 to make the coefficients of $x$ equal.
$3 \times (3x + 6y) = 3 \times 4$ $9x + 18y = 12 \quad (3')$
Subtract equation (1') from equation (3').
$(9x + 18y) - (9x + 12y) = 12 - (-3)$ $6y = 15$ $y = \frac{15}{6}$ $y = \frac{5}{2}$
Substitute the value of $y$ into equation (2') to solve for $x$ .
$3x + 6\left(\frac{5}{2}\right) = 4$ $3x + 3 \times 5 = 4$ $3x + 15 = 4$ $3x = 4 - 15$ $3x = -11$ $x = -\frac{11}{3}$
Convert back to $a$ and $b$ using $a = \frac{1}{x}$ and $b = \frac{1}{y}$ .
$a = \frac{1}{-\frac{11}{3}} = -\frac{3}{11}$ $b = \frac{1}{\frac{5}{2}} = \frac{2}{5}$
The values are $\boxed{a = -\frac{3}{11}, b = \frac{2}{5}}$ .

QUESTION 3: Solve for a and b in the equations below: a + 3/b = 4 a - 12/b = -3

Step-by-step solution

Evaluate the terms inside the parenthesis.

Substitute these values back into the expression.

Calculate the final value.

Apply the rule for exponents.

Find the number that, when multiplied by itself, equals 81.

Move the decimal point to the right until the number is between 1 and 10.

Count how many places the decimal point was moved.

Write the number in standard form ( $A \times 10^n$ ).

Start with the general quadratic equation and divide by $a$ (assuming $a \neq 0$ ).

Move the constant term to the right side of the equation.

Complete the square on the left side by adding $(\frac{b}{2a})^2$ to both sides.

Factor the left side as a perfect square and simplify the right side.

Take the square root of both sides.

Isolate $x$ to obtain the quadratic formula.

Let $x = \frac{1}{a}$ and $y = \frac{1}{b}$ to transform the system into linear equations.

Use the elimination method to solve for $x$ and $y$ .

Subtract equation (1') from equation (3').

Substitute the value of $y$ into equation (2') to solve for $x$ .

Convert back to $a$ and $b$ using $a = \frac{1}{x}$ and $b = \frac{1}{y}$ .

Still stuck on this one?

More Mathematics Questions

QUESTION 3: Solve for a and b in the equations below: a + 3/b = 4 a - 12/b = -3

Step-by-step solution

Evaluate the terms inside the parenthesis.

Substitute these values back into the expression.

Calculate the final value.

Apply the rule for exponents.

Find the number that, when multiplied by itself, equals 81.

Move the decimal point to the right until the number is between 1 and 10.

Count how many places the decimal point was moved.

Write the number in standard form (A×10nA \times 10^nA×10n).

Start with the general quadratic equation and divide by aaa (assuming a≠0a \neq 0a=0).

Move the constant term to the right side of the equation.

Complete the square on the left side by adding (b2a)2(\frac{b}{2a})^2(2ab​)2 to both sides.

Factor the left side as a perfect square and simplify the right side.

Take the square root of both sides.

Isolate xxx to obtain the quadratic formula.

Let x=1ax = \frac{1}{a}x=a1​ and y=1by = \frac{1}{b}y=b1​ to transform the system into linear equations.

Use the elimination method to solve for xxx and yyy.

Subtract equation (1') from equation (3').

Substitute the value of yyy into equation (2') to solve for xxx.

Convert back to aaa and bbb using a=1xa = \frac{1}{x}a=x1​ and b=1yb = \frac{1}{y}b=y1​.

Still stuck on this one?

More Mathematics Questions

Write the number in standard form ( $A \times 10^n$ ).

Start with the general quadratic equation and divide by $a$ (assuming $a \neq 0$ ).

Complete the square on the left side by adding $(\frac{b}{2a})^2$ to both sides.

Isolate $x$ to obtain the quadratic formula.

Let $x = \frac{1}{a}$ and $y = \frac{1}{b}$ to transform the system into linear equations.

Use the elimination method to solve for $x$ and $y$ .

Substitute the value of $y$ into equation (2') to solve for $x$ .

Convert back to $a$ and $b$ using $a = \frac{1}{x}$ and $b = \frac{1}{y}$ .