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Magazine Chapter 10: Problem 2
Fill in the blanks. We can solve \(x-2 \sqrt{x}-8=0\) by making the ____ \(y=\sqrt{x}\)
Short Answer
Expert verified
Make the substitution \(y = \sqrt{x}\).
Step by step solution
01
Identify the substitution
In the given equation \(x - 2\sqrt{x} - 8 = 0\), we are asked to make the substitution \(y = \sqrt{x}\). This substitution simplifies the equation by letting us express \(x\) in terms of \(y\).
02
Express \(x\) in terms of \(y\)
Since \(y = \sqrt{x}\), we have \(x = y^2\). This new expression will be used in the upcoming steps to transform the original equation.
03
Substitute into the equation
Replace \(x\) with \(y^2\) in the equation to form \(y^2 - 2y - 8 = 0\). Now, the equation is a quadratic in terms of \(y\).
04
Solve the quadratic equation
Solve the quadratic equation \(y^2 - 2y - 8 = 0\) using the quadratic formula: \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here, \(a = 1\), \(b = -2\), and \(c = -8\).
05
Calculate the discriminant
Calculate \(b^2 - 4ac = (-2)^2 - 4(1)(-8) = 4 + 32 = 36\). The discriminant is 36, which is a perfect square.
06
Find the roots
Use the discriminant to find the roots: \(y = \frac{2 \pm \sqrt{36}}{2} = \frac{2 \pm 6}{2}\). This gives two solutions: \(y = 4\) and \(y = -2\).
07
Interpret the solutions
Since \(y = \sqrt{x}\), \(y\) must be non-negative. Therefore, only \(y = 4\) is valid. Exclude \(y = -2\) as it does not satisfy the requirement of being non-negative.
08
Find the value of \(x\)
Using the relationship \(y = 4\) and \(y = \sqrt{x}\), solve for \(x = y^2 = 4^2 = 16\). Hence, the solution to the original equation is \(x = 16\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Substitution Method
The substitution method is a great technique to simplify complex equations, especially those involving square roots, such as the problem we are tackling. By substituting a complex part of an equation with a simpler variable, calculations become manageable. In our task, the substitution involves setting \( y = \sqrt{x} \).
- By letting \( y = \sqrt{x} \), we transform the original equation from one involving square roots into a simpler quadratic form \( y^2 - 2y - 8 = 0 \).
- This transformation helps to eliminate the square root, making it easier to handle using regular quadratic solving techniques.
Discriminant
The discriminant is a part of the quadratic formula that determines the nature of the roots of a quadratic equation. For any equation of the form \( ax^2 + bx + c = 0 \), the discriminant \( \Delta \) is given by the formula \( b^2 - 4ac \).
- If the discriminant is positive, like in our problem where \( 36 \) is our discriminant, the quadratic equation has two distinct real roots.
- If the discriminant equals zero, the equation has exactly one real root (also known as a repeated root).
- When the discriminant is negative, the quadratic equation has no real roots but rather two complex roots.
Quadratic Formula
The quadratic formula is a universal tool for solving any quadratic equation of the form \( ax^2 + bx + c = 0 \). The formula is given by:\[y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
This straightforward formula takes the coefficients \( a \), \( b \), and \( c \) from your equation to find the values of \( y \).
This straightforward formula takes the coefficients \( a \), \( b \), and \( c \) from your equation to find the values of \( y \).
- The \( -b \) part begins the process of finding the axis of symmetry for the roots.
- The \( \pm \sqrt{b^2 - 4ac} \) portion addresses the discriminant, letting us know if and how the roots differ.
- Finally, dividing by \( 2a \) ensures the results are centered around the vertex of the quadratic's parabola.
