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Magazine Chapter 5: Problem 3
Solve each equation by using the Square Root Property. \(x^{2}+16 x+64=7\)
Short Answer
Expert verified
The solutions are \(-8 + \sqrt{7}\) and \(-8 - \sqrt{7}\).
Step by step solution
01
Move Constant to the Other Side
First, move 64 to the right side of the equation by subtracting 64 from both sides. This gives you\[ x^2 + 16x = 7 - 64 \]which simplifies to\[ x^2 + 16x = -57 \]
02
Complete the Square
To complete the square, take half of the coefficient of \(x\), square it, and add it to both sides. Half of 16 is 8, and 8 squared is 64. Add 64 to both sides:\[ x^2 + 16x + 64 = -57 + 64 \]This becomes:\[ (x + 8)^2 = 7 \]
03
Apply the Square Root Property
Now, take the square root of both sides of the equation using the Square Root Property:\[ \sqrt{(x + 8)^2} = \pm \sqrt{7} \]This gives us:\[ x + 8 = \pm \sqrt{7} \]
04
Solve for x
Finally, isolate \(x\) by subtracting 8 from both sides:\[ x = -8 \pm \sqrt{7} \]So the solutions are:\[ x = -8 + \sqrt{7} \quad \text{and} \quad x = -8 - \sqrt{7} \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Completing the Square
Completing the square is a method used in algebra to simplify and solve quadratic equations. It involves creating a perfect square trinomial from a quadratic expression. But why do we want a perfect square? Because it lets us apply the square root property easily.
Here's how you complete the square:
Here's how you complete the square:
- First, ensure that the coefficient of the quadratic term is 1. If it's not, divide the entire equation by this coefficient.
- Next, look at the linear coefficient, the number in front of the \(x\) term.
- Divide this coefficient by 2, and then square the result. This squared number will be your magic number that you add to both sides of the equation.
- Now, rewrite the quadratic and linear terms along with your magic number as a perfect square trinomial, which can be expressed as \((x + a)^2\).
Quadratic Equations
Quadratic equations are polynomial equations of degree 2, which means the highest power of the variable is squared. They take the general form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. Solving quadratic equations can be approached in several ways.
Key methods include:
Key methods include:
- Factoring: This involves expressing the quadratic expression as a product of two binomials. This method works well when the equation can be factored easily.
- Completing the Square: This technique helps to rewrite the quadratic in a perfect square form, which then allows for straightforward solving using the square root.
- Quadratic Formula: Given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), this formula provides solutions directly, making it useful for complex or impossible-to-factor quadratics.
Algebraic Solutions
Algebraic solutions involve manipulating an equation using algebraic properties and operations to isolate the variable of interest. For quadratic equations, employing algebraic solutions typically involves re-arranging terms and leveraging mathematical properties.
To solve quadratics using algebra, we can:
To solve quadratics using algebra, we can:
- Move terms across the equal sign to isolate the quadratic, linear, and constant terms appropriately.
- Use operations such as addition, subtraction, multiplication, division, and the square root, to simplify and solve for \(x\).
- Follow systematic steps to maintain balance in the equation, such as completing the square or factoring.
- Cross-check the solutions by plugging back the values of \(x\) into the original equation to ensure they satisfy the equation.
