Question

day 1 - 9.5, 9.7, 9.10 - solve quadratics by factoring & square roots
student name:

1. solve the following equation:

0 = h² + 4h + 3

1. solve the following equation:

0 = k² + 9k + 18

1. solve the following equation:

766 = 10x² - 44

1. solve the following equation:

-18 = -7x² + 126
provide an answer accurate to the nearest hundredth.

Explanation:

Problem 1: Solve \( 0 = h^2 + 4h + 3 \)

Step 1: Factor the quadratic

We need two numbers that multiply to \( 3 \) and add to \( 4 \). Those numbers are \( 1 \) and \( 3 \). So, we can factor the quadratic as:
\( h^2 + 4h + 3 = (h + 1)(h + 3) \)

Step 2: Set each factor equal to zero

Using the zero - product property (if \( ab = 0 \), then \( a = 0 \) or \( b = 0 \)):
\( h + 1 = 0 \) or \( h + 3 = 0 \)

Step 3: Solve for \( h \)

For \( h + 1 = 0 \), subtract \( 1 \) from both sides: \( h=-1 \)
For \( h + 3 = 0 \), subtract \( 3 \) from both sides: \( h = - 3 \)

Step 1: Factor the quadratic

We need two numbers that multiply to \( 18 \) and add to \( 9 \). Those numbers are \( 3 \) and \( 6 \). So, we can factor the quadratic as:
\( k^2+9k + 18=(k + 3)(k + 6) \)

Step 2: Set each factor equal to zero

Using the zero - product property:
\( k + 3 = 0 \) or \( k + 6 = 0 \)

Step 3: Solve for \( k \)

For \( k + 3 = 0 \), subtract \( 3 \) from both sides: \( k=-3 \)
For \( k + 6 = 0 \), subtract \( 6 \) from both sides: \( k=-6 \)

Step 1: Isolate the \( x^2 \) term

First, add \( 44 \) to both sides of the equation:
\( 766+44=10x^2-44 + 44 \)
\( 810 = 10x^2 \)

Step 2: Solve for \( x^2 \)

Divide both sides by \( 10 \):
\( \frac{810}{10}=\frac{10x^2}{10} \)
\( 81=x^2 \)

Step 3: Solve for \( x \)

Take the square root of both sides. Remember that if \( x^2=a \) (\( a\geq0 \)), then \( x=\pm\sqrt{a} \)
\( x=\pm\sqrt{81}=\pm9 \)

Answer:

\( h=-1 \) or \( h = - 3 \)

Problem 2: Solve \( 0=k^2 + 9k + 18 \)