Question

solve the equation by factoring: $x^2 - 15 = -2x$
choose your answer...
14 multiple answer 2 points
a triangle has an area of 27 square inches. the height of the triangle is 1.5 times its base. find the base and height of the triangle.
base is 3 inches
height is 6 inches
base is 5 inches
height is 9 inches
base is 4 inches
height is 10 inches
base is 6 inches
height is 7 inches

Explanation:

First Problem: Solve the equation by factoring: \( x^2 - 15 = -2x \)

Step1: Rearrange the equation to standard quadratic form

We want to get all terms on one side so the equation is in the form \( ax^2 + bx + c = 0 \). Add \( 2x \) to both sides and add \( 15 \) to both sides:
\( x^2 + 2x - 15 = 0 \)

Step2: Factor the quadratic expression

We need two numbers that multiply to \( -15 \) (the product of \( a = 1 \) and \( c = -15 \)) and add up to \( 2 \) (the coefficient of \( x \)). The numbers \( 5 \) and \( -3 \) work because \( 5 \times (-3) = -15 \) and \( 5 + (-3) = 2 \). So we factor the quadratic as:
\( (x + 5)(x - 3) = 0 \)

Step3: Solve for \( x \) using the zero - product property

The zero - product property states that if \( ab = 0 \), then either \( a = 0 \) or \( b = 0 \). So we set each factor equal to zero:

• For \( x + 5 = 0 \), subtract \( 5 \) from both sides: \( x = - 5 \)
• For \( x - 3 = 0 \), add \( 3 \) to both sides: \( x = 3 \)

Second Problem: A triangle has an area of 27 square inches. The height of the triangle is 1.5 times its base. Find the base and height of the triangle.

The formula for the area of a triangle is \( A=\frac{1}{2}bh \), where \( A \) is the area, \( b \) is the base, and \( h \) is the height. We know that \( h = 1.5b \) and \( A = 27 \).

Step1: Substitute \( h = 1.5b \) into the area formula

Substitute \( h \) and \( A \) into \( A=\frac{1}{2}bh \):
\( 27=\frac{1}{2}b(1.5b) \)

Step2: Simplify the equation

First, multiply \( \frac{1}{2} \) and \( 1.5 \). \( \frac{1}{2}\times1.5 = 0.75 \), so the equation becomes:
\( 27 = 0.75b^2 \)

Step3: Solve for \( b^2 \)

Divide both sides by \( 0.75 \):
\( b^2=\frac{27}{0.75}=36 \)

Step4: Solve for \( b \)

Take the square root of both sides. Since length cannot be negative, we consider the positive square root:
\( b=\sqrt{36} = 6 \) inches

Step5: Find the height

Since \( h = 1.5b \), substitute \( b = 6 \):
\( h=1.5\times6 = 9 \) inches

First Problem Answer:

The solutions are \( x = - 5 \) and \( x = 3 \)

Second Problem Answer:

The correct options are:

• Base is 6 inches
• Height is 9 inches

Answer:

The formula for the area of a triangle is \( A=\frac{1}{2}bh \), where \( A \) is the area, \( b \) is the base, and \( h \) is the height. We know that \( h = 1.5b \) and \( A = 27 \).

Step1: Substitute \( h = 1.5b \) into the area formula

Substitute \( h \) and \( A \) into \( A=\frac{1}{2}bh \):
\( 27=\frac{1}{2}b(1.5b) \)

Step2: Simplify the equation

First, multiply \( \frac{1}{2} \) and \( 1.5 \). \( \frac{1}{2}\times1.5 = 0.75 \), so the equation becomes:
\( 27 = 0.75b^2 \)

Step3: Solve for \( b^2 \)

Divide both sides by \( 0.75 \):
\( b^2=\frac{27}{0.75}=36 \)

Step4: Solve for \( b \)

Take the square root of both sides. Since length cannot be negative, we consider the positive square root:
\( b=\sqrt{36} = 6 \) inches

Step5: Find the height

Since \( h = 1.5b \), substitute \( b = 6 \):
\( h=1.5\times6 = 9 \) inches

First Problem Answer:

The solutions are \( x = - 5 \) and \( x = 3 \)

Second Problem Answer:

The correct options are:

• Base is 6 inches
• Height is 9 inches