Use the quadratic formula x = −b +/- sqrt(b² − 4ac)2a. Here, a=2, b=−13, c=20.

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1. Solve the equation $2x^2 - 13x + 20 = 0$ .

Use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .
Here, $a=2$ , $b=-13$ , $c=20$ .
$x = \frac{-(-13) \pm \sqrt{(-13)^2 - 4(2)(20)}}{2(2)}$
Simplify the expression under the square root.
$x = \frac{13 \pm \sqrt{169 - 160}}{4}$ $x = \frac{13 \pm \sqrt{9}}{4}$
Calculate the square root.
$x = \frac{13 \pm 3}{4}$
Find the two possible values for $x$ .
$x_1 = \frac{13 + 3}{4} = \frac{16}{4} = 4$ $x_2 = \frac{13 - 3}{4} = \frac{10}{4} = \frac{5}{2}$
The solutions are:
$\boxed{x = 4 \text{ or } x = \frac{5}{2}}$
2. A delivery company models fuel consumption using the rule: $F = \frac{64a^7b^9}{8a^3b^3}$ . Simplify using index laws.
Separate the numerical coefficients and the variables.
$F = \left(\frac{64}{8}\right) \times \left(\frac{a^7}{a^3}\right) \times \left(\frac{b^9}{b^3}\right)$
Simplify the numerical coefficient and apply the index law $\frac{x^m}{x^n} = x^{m-n}$ for the variables.
$F = 8 \times a^{7-3} \times b^{9-3}$
Perform the subtractions in the exponents.
$F = \boxed{8a^4b^6}$
3. If $a=2$ and $b=3$ , find the numerical value of the simplified expression from .
Substitute $a=2$ and $b=3$ into the simplified expression $F = 8a^4b^6$ .
$F = 8(2)^4(3)^6$
Calculate the powers.
$F = 8(16)(729)$
Multiply the values.
$F = 128 \times 729$ $F = \boxed{93312}$
4. A student plans to fence a rectangular kitchen garden whose area is given by $A = x^2 - 4x - 21$ . Factorize.
Find two numbers that multiply to -21 and add up to -4.
These numbers are 3 and -7.
Rewrite the quadratic expression using these numbers.
$A = x^2 + 3x - 7x - 21$
Factor by grouping.
$A = x(x + 3) - 7(x + 3)$
Factor out the common binomial factor.
$A = \boxed{(x + 3)(x - 7)}$

Use the quadratic formula x = -b +/- sqrt(b² - 4ac)2a. Here, a=2, b=-13, c=20.

Step-by-step solution

Use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .

Simplify the expression under the square root.

Calculate the square root.

Find the two possible values for $x$ .

Separate the numerical coefficients and the variables.

Simplify the numerical coefficient and apply the index law $\frac{x^m}{x^n} = x^{m-n}$ for the variables.

Perform the subtractions in the exponents.

Substitute $a=2$ and $b=3$ into the simplified expression $F = 8a^4b^6$ .

Calculate the powers.

Multiply the values.

Find two numbers that multiply to -21 and add up to -4.

Rewrite the quadratic expression using these numbers.

Factor by grouping.

Factor out the common binomial factor.

Still stuck on this one?

More Mathematics Questions

Use the quadratic formula x = -b +/- sqrt(b² - 4ac)2a. Here, a=2, b=-13, c=20.

Step-by-step solution

Use the quadratic formula x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac​​.

Simplify the expression under the square root.

Calculate the square root.

Find the two possible values for xxx.

Separate the numerical coefficients and the variables.

Simplify the numerical coefficient and apply the index law xmxn=xm−n\frac{x^m}{x^n} = x^{m-n}xnxm​=xm−n for the variables.

Perform the subtractions in the exponents.

Substitute a=2a=2a=2 and b=3b=3b=3 into the simplified expression F=8a4b6F = 8a^4b^6F=8a4b6.

Calculate the powers.

Multiply the values.

Find two numbers that multiply to -21 and add up to -4.

Rewrite the quadratic expression using these numbers.

Factor by grouping.

Factor out the common binomial factor.

Still stuck on this one?

More Mathematics Questions

Use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .

Find the two possible values for $x$ .

Simplify the numerical coefficient and apply the index law $\frac{x^m}{x^n} = x^{m-n}$ for the variables.

Substitute $a=2$ and $b=3$ into the simplified expression $F = 8a^4b^6$ .