10.8: Comparing Methods for Solving Quadratics
Suppose you need to solve the quadratic equation \begin{align*}-16t^2+22t+3=0\end{align*} in order to determine after how many seconds a pebble shot up into the air by a slingshot will hit the ground. How could you solve such an equation? Could you use graphing, factoring, taking the square root, completing the square, and/or the quadratic formula? How would you choose which method to use? After completing this Concept, you'll be able to choose the best method for solving a quadratic problem such as this one.
Guidance
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 with 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 A
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.
In order to find the stopping distance \begin{align*}s\end{align*} of a car with a constant deceleration \begin{align*}a\end{align*} applied when at a speed \begin{align*}u\end{align*}, this formula can be used: \begin{align*} s = \frac{u^2}{2a}.\end{align*}
Example B
Find the maximum speed of a car if it must stop before 30 feet and if the constant deceleration is 20 feet per second squared.
Solution:
This problem can be solved by substituting, simplifying, and taking the square root:
\begin{align*} \text{Begin with the equation:} && s &= \frac{u^2}{2a}\\ \text{Substitute in the values:} && 30 &= \frac{u^2}{2\cdot 20} \\ \text{Simplify algebraically:} && 1200 &= u^2\\ \text{Take the square root of each side:} && u &\approx \pm 34.64 \end{align*}
Since negative speed does not make sense, the maximum speed is 34.64 feet per second.
Example C
The product of two consecutive even integers is 288. Find the integers.
Solution:
Let the first integer be \begin{align*}n\end{align*}. Then the next even integer is 2 larger: \begin{align*}n+2\end{align*}. Their product is:
\begin{align*}n(n+2)=288 \Rightarrow n^2+2n-288=0\end{align*}
This can be solved by factoring or using the quadratic equation. Either way, \begin{align*}n=72\end{align*}, and so the two numbers are 72 and 74.
Guided Practice
The Bernoulli effect says that there is a relationship between the speed \begin{align*}u\end{align*} and the pressure \begin{align*}P\end{align*} of an air particle at a given height \begin{align*}h\end{align*}:
\begin{align*}\frac{u^2}{2} + P = h\end{align*}
Find the speed of an air particle at a height of 1000 meters with a pressure of 978.25 millibars.
Solution:
This problem can be solved by substituting, simplifying and taking the square root:
\begin{align*} \text{Begin with the equation:} && \frac{u^2}{2} + P &= h\\ \text{Substitute in the values:} && \frac{u^2}{2} + 978.25 &= 1000 \\ \text{Simplify algebraically:} && u^2 &= 43.5\\ \text{Take the square root of each side:} && u &\approx \pm 6.6 \end{align*}
Negative speed does not make sense in this context, and so the speed is 6.6 meters per second.
Practice
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 to 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
- A theater 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 theater has a total of 1,100 seats. Determine the number of balcony, box, and floor seats in the theater.
- Write an equation in slope-intercept form containing (10, 65) and (5, 30).
- 120% of what number is 60?
- Name the set(s) 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.
solutions of a quadratic equation
The intercept of a quadratic equation is also called a root, solution, or zero.Perfect Square Trinomial
A perfect square trinomial is a quadratic expression of the form (which can be rewritten as ) or (which can be rewritten as ).Quadratic Formula
The quadratic formula states that for any quadratic equation in the form , .Image Attributions
Here you'll learn how to choose between the different techniques for solving quadratic problems learned in previous Concepts.
Concept Nodes:
solutions of a quadratic equation
The intercept of a quadratic equation is also called a root, solution, or zero.Perfect Square Trinomial
A perfect square trinomial is a quadratic expression of the form (which can be rewritten as ) or (which can be rewritten as ).Quadratic Formula
The quadratic formula states that for any quadratic equation in the form , .