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Chapter 0: Problem 105

Solve each quadratic equation by the method of your choice. $$4 x^{2}-16=0$$

Short Answer

Expert verified
The solutions for the quadratic equation \(4x^{2} - 16 = 0\) are \(x = 2\) and \(x = -2\).

Step by step solution

01

Rewrite the equation

The first step is to rewrite the equation in the form \(x^2 = k\), where \(k\) is a constant term. Divide both sides by 4, yielding: \(x^{2} - 4 = 0\).
02

Isolate \(x^2\)

Next, isolate \(x^2\) from the equation. Add 4 to both sides of the equation yielding: \(x^{2} = 4\).
03

Solve for \(x\)

Now, apply square root to both sides: \(x = \pm \sqrt{4}\). Which simplifies to \(x = \pm 2\).

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Key Concepts

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

Quadratic Equation
When we talk about a quadratic equation, we refer to a second-degree polynomial equation of the general form \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are constants, with \(a\) being non-zero. Solving a quadratic equation means finding the values of \(x\) that make the equation true, known as the equation's roots.

There are various methods to solve these equations, such as factoring, completing the square, using the quadratic formula, or graphing. The choice of method often depends on the specific form of the quadratic equation at hand and the preferences of the person solving it.
Factoring Quadratics
The technique of factoring quadratics involves rewriting the quadratic equation in a product of two binomials. We do this by looking for two numbers that multiply to get \(ac\) (where \(a\) is the coefficient of \(x^2\) and \(c\) is the constant term) and add to get \(b\) (the coefficient of \(x\)).

If we can find such numbers, we can then express the quadratic as \( (x + m)(x + n) = 0\), where \(m\) and \(n\) are the numbers we found. This form shows us that the solutions to the quadratic equation are the values that make each binomial equal to zero, known as the zero product property. Factoring is most efficient when the equation can be easily factorable into integer or rational numbers.
Square Root Method
The square root method is a direct approach for solving quadratics of the form \(x^2 = k\), where \(k\) is a constant. To apply this method, we take the following steps:
  • Ensure the equation is in the form \(x^2 = k\) by performing necessary operations to isolate the \(x^2\) term.
  • Apply the square root to both sides of the equation which gives us \(x = \(\pm\sqrt{k}\).
  • Since squaring a number results in a positive value, \(x\) can have both a positive and negative root hence, the \(\pm\) symbol.

The solution set will consist of two numbers (unless \(k\) is zero, in which case the solution is \x = 0\)). When \(k\) is negative, this indicates that the equation has no real solutions since the square root of a negative number is not real in the set of real numbers.

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Most popular questions from this chapter

Perform the indicated operations. $$\left(1-\frac{1}{x}\right)\left(1-\frac{1}{x+1}\right)\left(1-\frac{1}{x+2}\right)\left(1-\frac{1}{x+3}\right)$$

What is a compound inequality and how is it solved?

Place the correct symbol, \(>\) or \(<\), in the shaded area between the given numbers. Do not use a calculator. Then check your result with a calculator. \(a, 3^{\frac{1}{2}}\square 3^{1}\) b. \(\sqrt{7}+\sqrt{18} \square \sqrt{7+18}\)

Explain how to determine the restrictions on the variable for the equation $$ \frac{3}{x+5}+\frac{4}{x-2}=\frac{7}{x^{2}+3 x-6} $$

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. The toll to a bridge is \(\$ 3.00 .\) A three-month pass costs \(\$ 7.50\) and reduces the toll to \(\$ 0.50 .\) A six-month pass costs \(\$ 30\) and permits crossing the bridge for no additional fee. How many crossings per three-month period does it take for the three-month pass to be the best deal?
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