Solving Equations#
How to Solve an Equation#
Steps for Solving an Equation
To find all values of \(x\) that satisfy an equation (e.g., \(f(x) = g(x)\)), complete the following steps:
Rewrite equation as \(f(x) - g(x) = 0\), if needed.
Simplify and factor the left-hand side.
Set each factor equal to zero and solve for \(x\).
Example 1#
Solving a quadratic equation
Find all values of \(x\) such that \(x^2 - 4x - 12 = 0\).
Step 1: Factor \(x^2 - 4x - 12\).
Recall from Factoring, Example 8 that
Step 2: Set each factor equal to zero and solve for \(x\).
Therefore, \(x=-2\) and \(x=6\) satisfy \(x^2 - 4x - 12 = 0\).
Check Our Work.
We can check our work by plugging in \(x=-2\) and \(x=6\) into the given polynomial and making sure it evaluates to zero.
Video Resource

Solving Quadratic Equations by Factoring (Links to an external site)
Several more examples of solving quadratic equations.
Example 2#
Solving a rational equation
Find all values of \(p\) such that \(\dfrac{3p}{180-6p} = 1\).
Step 1: Multiply both sides of \(\frac{3p}{180-6p} = 1\) by the denominator, \(180-6p\).
Step 2: Subtract \(180 - 6p\) from both sides and simplify.
Step 3: Solve for \(p\).
Therefore, \(p=20\) is the only value that satisfies \(\frac{3p}{180-6p} = 1\).
Check Our Work.
We can check our work by plugging in \(p=20\) into \(\dfrac{3p}{180-6p}\) and making sure it evaluates to one.
Example 3#
Points of intersection
Find all points of intersection of \(f(x) = 6x^2 - 4x\) and \(g(x) = 2 - 5x\).
Step 1: Set \(f(x) = g(x)\).
Points of intersection can be found by setting the two curves equal to each other and solving for \(x\).
Step 2: Subtract \(2-5x\) from both sides of the equation in Step 1 and simplify.
Step 3: Use the AC grouping method to factor \(6x^2 + x - 2\).
Find two integers that multiply to \(6(-2) = -12\) and sum to \(1\).
Product equals \(-12\) |
Sum equals \(1\)? |
---|---|
\(-1 \times 12\) |
NO |
\(-2 \times 6\) |
NO |
\(-3 \times 4\) |
YES |
Therefore,
Step 4: Set each factor equal to zero and solve for \(x\).
Therefore, the only points of intersection of \(f(x) = 6x^2 - 4x\) and \(g(x) = 2 - 5x\) occur when \(x=-2/3\) and \(x=1/2\).
Check Our Work.
We can check our work by making sure \(f(x) = g(x)\) when we plug in \(x=-2/3\) and when we plug int \(x=1/2\).
\(x\) |
\(f(x) = 6x^2 - 4x\) |
\(g(x) = 2 - 5x\) |
---|---|---|
\(-2/3\) |
\(6(-2/3)^2 - 4(-2/3) = 16/3 \) |
\(2 - 5(-2/3) = 16/3\) |
\(1/2\) |
\(6(1/2)^2 - 4(1/2) = -1/2\) |
\(2 - 5(1/2) = -1/2\) |