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Magazine Chapter 5: Problem 24
Solve each equation, and check the solutions. $$ p^{2}+8 p+7=0 $$
Short Answer
Expert verified
-1 and -7
Step by step solution
01
Identify the form of the quadratic equation
Recognize that the given equation is a quadratic equation in the form of \( ax^2 + bx + c = 0 \). Here, we have \( a = 1 \), \( b = 8 \), and \( c = 7 \).
02
Write the quadratic formula
To solve the quadratic equation, use the quadratic formula: \[ p = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}. \]
03
Substitute the values into the quadratic formula
Substitute \( a = 1 \), \( b = 8 \), and \( c = 7 \) into the quadratic formula: \[ p = \frac{{-8 \pm \sqrt{{8^2 - 4(1)(7)}}}}{2(1)} \]
04
Simplify under the square root
Calculate the value under the square root:\( 8^2 - 4(1)(7) = 64 - 28 = 36 \). So we have:\[ p = \frac{{-8 \pm \sqrt{36}}}{2} \]
05
Solve for the roots
Now solve for the two possible values of \( p \):\[ p = \frac{{-8 + 6}}{2} = \frac{-2}{2} = -1 \]and\[ p = \frac{{-8 - 6}}{2} = \frac{-14}{2} = -7 \]
06
Check the solutions
Substitute \( p = -1 \) into the original equation:\[ (-1)^2 + 8(-1) + 7 = 1 - 8 + 7 = 0 \]. So \( p = -1 \) is a solution.Substitute \( p = -7 \) into the original equation:\[ (-7)^2 + 8(-7) + 7 = 49 - 56 + 7 = 0 \]. So \( p = -7 \) is a solution.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding the Quadratic Formula
Quadratic equations are equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. The quadratic formula is a fundamental tool to solve these equations. The formula is given by: \[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \]This formula finds the values of \x\ (or the roots) that satisfy the quadratic equation. The term under the square root \(\sqrt{{b^2 - 4ac}}\), is called the discriminant. It helps to determine the nature of the roots:
Here, we applied it to \(p^2 + 8p + 7 = 0\) to find its solutions.
- If \(b^2 - 4ac > 0\), there are two distinct real roots.
- If \(b^2 - 4ac = 0\), there is exactly one real root.
- If \(b^2 - 4ac < 0\), there are no real roots, only complex ones.
Here, we applied it to \(p^2 + 8p + 7 = 0\) to find its solutions.
Finding the Roots of the Quadratic Equation
Roots are values of the variable that satisfy the equation \(ax^2 + bx + c = 0\). To find these roots using the quadratic formula:
First, identify the coefficients from the equation. For \(p^2 + 8p + 7 = 0\), we have:
First, identify the coefficients from the equation. For \(p^2 + 8p + 7 = 0\), we have:
- \(a = 1 \)
- \(b = 8 \)
- \(c = 7 \)
- \( p = \frac{{-8 + \sqrt{36}}}{2} = \frac{-2}{2} = -1 \)
- \( p = \frac{{-8 - \sqrt{36}}}{2} = \frac{-14}{2} = -7 \)
Checking the Solutions
After solving a quadratic equation, it's crucial to verify your solutions by substituting them back into the original equation. This ensures there were no errors in the calculations.
For \(p = -1\):
\ ((-1)^2 + 8(-1) + 7 = 1 - 8 + 7 = 0) \ confirms that \ p = -1\ is a valid solution.
For \(p = -7\):
\ ((-7)^2 + 8(-7) + 7 = 49 - 56 + 7 = 0) \ confirms that \ p = -7\ is a valid solution.
By checking the solutions this way, you can be confident that your answers are correct. Always remember to substitute back into the original equation for verification.
For \(p = -1\):
\ ((-1)^2 + 8(-1) + 7 = 1 - 8 + 7 = 0) \ confirms that \ p = -1\ is a valid solution.
For \(p = -7\):
\ ((-7)^2 + 8(-7) + 7 = 49 - 56 + 7 = 0) \ confirms that \ p = -7\ is a valid solution.
By checking the solutions this way, you can be confident that your answers are correct. Always remember to substitute back into the original equation for verification.
