College Algebra: Solving Quadratic Equations by Factoring

Q1. Solve by factoring: $x^2 - 7x - 18 = 0$

Background

Topic: Quadratic Equations and Factoring

This question tests your ability to solve quadratic equations by factoring. Factoring is a method used to rewrite a quadratic equation as a product of two binomials set equal to zero, allowing you to find the solutions (roots) of the equation.

Key Terms and Formulas

• Quadratic Equation: $ax^2 + bx + c = 0$

• Factoring: Rewriting an expression as a product of its factors

• Zero Product Property: If $ab = 0$, then $a = 0$ or $b = 0$

Step-by-Step Guidance

1. Write the equation in standard quadratic form: $x^2 - 7x - 18 = 0$.

2. Identify $a = 1$, $b = -7$, and $c = -18$.

3. Look for two numbers that multiply to $-18$ (the constant term) and add to $-7$ (the coefficient of $x$).

4. Express the quadratic as a product of two binomials: $(x + m)(x + n) = 0$, where $m$ and $n$ are the numbers you found.

Try solving on your own before revealing the answer!

Q2. Solve by factoring: $4x^2 = 36x$

Background

Topic: Quadratic Equations and Factoring

This question asks you to solve a quadratic equation by factoring. You will need to rearrange the equation into standard form before factoring.

Key Terms and Formulas

• Quadratic Equation: $ax^2 + bx + c = 0$

• Factoring out the Greatest Common Factor (GCF)

• Zero Product Property

Step-by-Step Guidance

1. Move all terms to one side to set the equation to zero: $4x^2 - 36x = 0$.

2. Factor out the greatest common factor from both terms.

3. Set each factor equal to zero using the zero product property.

4. Solve each resulting equation for $x$.

Try solving on your own before revealing the answer!

Q3. Solve by factoring: $5x^2 = 25$

Background

Topic: Quadratic Equations and Factoring

This question involves solving a quadratic equation by first moving all terms to one side and then factoring.

Key Terms and Formulas

• Quadratic Equation: $ax^2 + bx + c = 0$

• Factoring

• Zero Product Property

Step-by-Step Guidance

1. Subtract $25$from both sides to write the equation in standard form: $5x^2 - 25 = 0$.

2. Factor out the greatest common factor from the equation.

3. Set each factor equal to zero using the zero product property.

4. Solve for $x$ in each resulting equation.

Try solving on your own before revealing the answer!