10.5: Solving Quadratic Equations Using the Quadratic Formula
This chapter has presented three methods to solve a quadratic equation:
- By graphing to find the zeros;
- By solving using square roots; and
- By using completing the square to find the solutions
This lesson will present a fourth way to solve a quadratic equation: using the Quadratic Formula.
History of the Quadratic Formula
As early as 1200 BC, people were interested in solving quadratic equations. The Babylonians solved simultaneous equations involving quadratics. In 628 AD, Brahmagupta, an Indian mathematician, gave the first explicit formula to solve a quadratic equation. The Quadratic Formula was written as it is today by the Arabic mathematician Al-Khwarizmi. It is his name upon which the word “Algebra” is based.
The solution to any quadratic equation in standard form \begin{align*}0=ax^2+bx+c\end{align*} is
\begin{align*}x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}\end{align*}
Example: Solve \begin{align*}x^2+10x+9=0\end{align*} using the Quadratic Formula.
Solution: We know from the last lesson the answers are \begin{align*}x=-1\end{align*} or \begin{align*}x=-9\end{align*}.
By applying the Quadratic Formula and \begin{align*}a=1, b=10\end{align*}, and \begin{align*}c=9\end{align*}, we get:
\begin{align*}x &= \frac{-10 \pm \sqrt{(10)^2-4(1)(9)}}{2(1)}\\ x &= \frac{-10 \pm \sqrt{100-36}}{2}\\ x &= \frac{-10 \pm \sqrt{64}}{2}\\ x &= \frac{-10 \pm 8}{2}\\ x &= \frac{-10 + 8}{2} \ or \ x=\frac{-10-8}{2}\\ x &= -1 \ or \ x=-9\end{align*}
Example 1: Solve \begin{align*}-4x^2+x+1=0\end{align*} using the Quadratic Formula.
Solution:
\begin{align*}\text{Quadratic formula:} && x & =\frac{-b \pm \sqrt{b^2-4ac}}{2a}\\ \text{Plug in the values} \ a=-4, b=1, c=1. && x& =\frac{-1 \pm \sqrt{(1)^2-4(-4)(1)}}{2(-4)}\\ \text{Simplify.} && x & =\frac{-1 \pm \sqrt{1+16}}{-8}=\frac{-1 \pm \sqrt{17}}{-8}\\ \text{Separate the two options.} && x&=\frac{-1+\sqrt{17}}{-8} \ \text{and} \ x=\frac{-1-\sqrt{17}}{-8}\\ \text{Solve.} && x & \approx -.39 \ \text{and} \ x \approx .64\end{align*}
Multimedia Link For more examples of solving quadratic equations using the Quadratic Formula, see Khan Academy Equation Part 2 (9:14).
Figure 2 provides more examples of solving equations using the quadratic equation. This video is not necessarily different from the examples above, but it does help reinforce the procedure of using the Quadratic Formula to solve equations.
Finding the Vertex of a Quadratic Equation in Standard Form
The \begin{align*}x-\end{align*}coordinate of the vertex of \begin{align*}0=ax^2+bx+c\end{align*} is \begin{align*}x=-\frac{b}{a}\end{align*}
Which Method to Use?
Usually you will not be told which method to use. You will have to make that decision yourself. However, here are some guidelines to which methods are better in different situations.
- Graphing – a good method to visualize the parabola and easily see the intersections. Not always precise.
- Factoring – best if the quadratic expression is easily factorable
- Taking the square root – is best used of the form \begin{align*}0=ax^2-c\end{align*}
- Completing the square – can be used to solve any quadratic equation. It is a very important method for rewriting a quadratic function in vertex form.
- Quadratic Formula – is the method that is used most often for solving a quadratic equation. If you are using factoring or the Quadratic Formula, make sure that the equation is in standard form.
Example: The length of a rectangular pool is 10 meters more than its width. The area of the pool is 875 square meters. Find the dimensions of the pool.
Solution: Begin by drawing a sketch. The formula for the area of a rectangle is \begin{align*}A=l(w)\end{align*}.
\begin{align*}A &= (x+10)(x)\\ 875 &= x^2+10x\end{align*}
Now solve for \begin{align*}x\end{align*} using any method you prefer.
The result is \begin{align*}x=25\end{align*}. So, the length of the pool is 35 meters and the width is 25 meters.
Practice Set
The following video will guide you through a proof of the Quadratic Formula. CK-12 Basic Algebra: Proof of Quadratic Formula (7:44)
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Using the Quadratic Formula (16:32)
- What is the Quadratic Formula? When is the most appropriate situation to use this formula?
- When was the first known solution of a quadratic equation recorded?
Find the \begin{align*}x-\end{align*}coordinate of the vertex of the following equations.
- \begin{align*}x^2-14x+45=0\end{align*}
- \begin{align*}8x^2-16x-42=0\end{align*}
- \begin{align*}4x^2+16x+12=0\end{align*}
- \begin{align*}x^2+2x-15=0\end{align*}
Solve the following quadratic equations using the Quadratic Formula.
- \begin{align*}x^2+4x-21=0\end{align*}
- \begin{align*}x^2-6x=12\end{align*}
- \begin{align*}3x^2-\frac{1}{2}x=\frac{3}{8}\end{align*}
- \begin{align*}2x^2+x-3=0\end{align*}
- \begin{align*}-x^2-7x+12=0\end{align*}
- \begin{align*}-3x^2+5x=0\end{align*}
- \begin{align*}4x^2=0\end{align*}
- \begin{align*}x^2+2x+6=0\end{align*}
Solve the following quadratic equations using the method of your choice.
- \begin{align*}x^2-x=6\end{align*}
- \begin{align*}x^2-12=0\end{align*}
- \begin{align*}-2x^2+5x-3=0\end{align*}
- \begin{align*}x^2+7x-18=0\end{align*}
- \begin{align*}3x^2+6x=-10\end{align*}
- \begin{align*}-4x^2+4000x=0\end{align*}
- \begin{align*}-3x^2+12x+1=0\end{align*}
- \begin{align*}x^2+6x+9=0\end{align*}
- \begin{align*}81x^2+1=0\end{align*}
- \begin{align*}-4x^2+4x=9\end{align*}
- \begin{align*}36x^2-21=0\end{align*}
- \begin{align*}x^2+2x-3=0\end{align*}
- The product of two consecutive integers is 72. Find the two numbers.
- The product of two consecutive odd integers is 11 less than 3 times their sum. Find the integers.
- The length of a rectangle exceeds its width by 3 inches. The area of the rectangle is 70 square inches. Find its dimensions.
- Suzie wants to build a garden that has three separate rectangular sections. She wants to fence around the whole garden and between each section as shown. The plot is twice as long as it is wide and the total area is 200 square feet. How much fencing does Suzie need?
- Angel wants to cut off a square piece from the corner of a rectangular piece of plywood. The larger piece of wood is \begin{align*}4 \ \text{feet} \times 8 \ \text{feet}\end{align*} and the cut off part is \begin{align*}\frac{1}{3}\end{align*} of the total area of the plywood sheet. What is the length of the side of the square?
- Mike wants to fence three sides of a rectangular patio that is adjacent the back of his house. The area of the patio is \begin{align*}192 \ ft^2\end{align*} and the length is 4 feet longer than the width. Find how much fencing Mike will need.
Mixed Review
- The theatre has three types of seating: balcony, box, and floor. There are four times as many floor seats as balcony. There are 200 more box seats than balcony seats. The theatre has a total of 1,100 seats. Determine the number of balcony, box, and floor seats in the theatre.
- Write an equation in slope-intercept form containing (10, 65) and (5, 30).
- 120% of what number is 60?
- Name the set() of numbers to which \begin{align*}\sqrt{16}\end{align*} belongs.
- Divide \begin{align*}6 \frac{1}{7} \div - 2 \frac{3}{4}\end{align*}.
- The set is the number of books in a library. Which of the following is the most appropriate domain for this set: all real numbers; positive real numbers; integers; or whole numbers? Explain your reasoning.