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Chapter 11: Problem 49

Solve. \(p^{4}-p^{2}-20=0\)

Short Answer

Expert verified
The solutions for \(p\) are \(p = \pm\sqrt{5}\).

Step by step solution

01

Identify substitution

The expression is a quadratic in terms of \(p^2\). Let's use the substitution \(x = p^2\). This gives us \(x^2 = (p^2)^2 = p^4\).
02

Substitute and rewrite the equation

Substitute \(x = p^2\) into the equation: \(x^2 - x - 20 = 0\). Now, we need to solve this quadratic equation for \(x\).
03

Solve the quadratic

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\) where \(a = 1\), \(b = -1\), and \(c = -20\). Substitute to get \(x = \frac{1 \pm \sqrt{1 + 80}}{2} = \frac{1 \pm \sqrt{81}}{2}\).
04

Calculate the roots of the quadratic

Simplify \(\sqrt{81}\) which equals 9. Therefore, the roots are \(x = \frac{1 + 9}{2} = 5\) and \(x = \frac{1 - 9}{2} = -4\).
05

Reverse substitution

Recall substitution \(x = p^2\). Hence, \(p^2 = 5\) or \(p^2 = -4\).
06

Solve for \(p\)

For \(p^2 = 5\), \(p = \pm\sqrt{5}\). For \(p^2 = -4\), there are no real solutions since the square of a real number cannot be negative.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

quadratic formula
The quadratic formula is a powerful tool for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). It's given by the formula:

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
This formula allows you to find the values of \(x\) that satisfy the equation. Here's how it works:
  • a, b, and c are coefficients from your equation.
  • The expression under the square root, \(b^2 - 4ac\), is called the discriminant.
  • The discriminant tells you about the nature of the roots鈥攚hether they are real or complex.
In the context of our problem, we used \(a = 1\), \(b = -1\), and \(c = -20\). Substituting these values into the quadratic formula, we simplified and determined possible roots for \(x\). Using the quadratic formula ensures a straightforward method that isn't dependent on factoring or guesswork, which is especially useful for complex equations.
reverse substitution
Reverse substitution is used to revert back to the original variable after solving a transformed equation. In our problem, we first substituted \(p^2 = x\) to simplify the equation from a quartic to a quadratic form. Here's how reverse substitution comes into play:
  • Initially, transform \(p^4 - p^2 - 20 = 0\) into a simpler quadratic form: \(x^2 - x - 20 = 0\).
  • After solving \(x^2 - x - 20 = 0\) and finding \(x = 5\) or \(x = -4\), replace \(x\) again with \(p^2\) to determine the values of \(p\).
The concept emphasizes that substitution and reverse substitution are techniques best paired together. Solve the modified equation, then translate the solutions back to the context of your original problem. This method simplifies complex equations, providing a manageable path toward finding solutions.
real solutions
Real solutions in the context of quadratic equations refer to the solutions where the square root component in the quadratic formula yields a real number. For a solution to be real, the discriminant \(b^2 - 4ac\) in the quadratic formula must be greater than or equal to zero:
  • If \( b^2 - 4ac > 0\), there are two distinct real solutions.
  • If \( b^2 - 4ac = 0\), there is one real solution (a repeated root).
  • If \( b^2 - 4ac < 0\), the solutions are complex and not real.
In our problem, when we computed \(x = 5\), it presented real solutions since it's capable of being expressed in terms of real numbers as \(p = \pm\sqrt{5}\). However, when considering \(x = -4\), \(p^2 = -4\) offers no real solutions since a square of a real number cannot be negative, thus affirming the absence of real solutions in this case.

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Most popular questions from this chapter

Sketch the graph of each quadratic function. Label the vertex, and sketch and label the axis of symmetry. See Examples I through 5 . $$ f(x)=(x-2)^{2}+5 $$

Sketch the graph of each quadratic function. Label the vertex, and sketch and label the axis of symmetry. $$ h(x)=10(x+4)^{2}-6 $$

Sketch the graph of each quadratic function. Label the vertex, and sketch and label the axis of symmetry. See Example 8 . $$ g(x)=4(x-4)^{2}+2 $$

Sketch the graph of each quadratic function. Label the vertex, and sketch and label the axis of symmetry. See Examples I through 5 . $$ H(x)=(x-1)^{2} $$

Sketch the graph of each quadratic function. Label the vertex, and sketch and label the axis of symmetry. $$ G(x)=\sqrt{5}(x-7)^{2}-\frac{1}{2} $$
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