Exercises 137–139 will help you prepare for the material covered in the next section. Solve: x(x - 7) = 3.

Verified step by step guidance

1. Start by expanding the left side of the equation: $x(x-7)=x\cdot x-x\cdot 7=x^2-\; 7x$.
2. Rewrite the equation with the expanded form: $x^2-\; 7x\; =\; 3$.
3. Bring all terms to one side to set the equation equal to zero: $x^2-\; 7x\; -\; 3\; =\; 0$.
4. Identify the coefficients for the quadratic equation in standard form $ax^2\; +\; bx\; +\; c\; =\; 0$: here, $a\; =\; 1$, $b\; =\; -7$, and $c\; =\; -3$.
5. Use the quadratic formula to solve for $x$: $x\; =\; \backslash frac\{-b\; \backslash pm\; \backslash sqrt\{b^2\; -\; 4ac\}\}\{2a\}$. Substitute the values of $a$, $b$, and $c$ into the formula.

Key Concepts

Expanding and Simplifying Algebraic Expressions

This involves applying the distributive property to multiply terms within parentheses and then combining like terms. For example, expanding x(x - 7) results in x² - 7x, which simplifies the equation for easier manipulation.

Solving Quadratic Equations

Quadratic equations are polynomial equations of degree two, typically in the form ax² + bx + c = 0. Solving them can involve factoring, completing the square, or using the quadratic formula to find the values of x that satisfy the equation.

Setting Equations to Zero

To solve quadratic equations, it is essential to rewrite the equation so that one side equals zero. This allows the use of factoring or the quadratic formula by isolating all terms on one side, creating a standard form ax² + bx + c = 0.