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Magazine Chapter 1: Problem 6
Solve the equation by factoring. \(4 x^{2}-4 x-15=0\)
Short Answer
Expert verified
The solutions are \(x = \frac{5}{2}\) and \(x = -\frac{3}{2}\).
Step by step solution
01
Identify the equation to factor
The equation we have is \(4x^2 - 4x - 15 = 0\). This is a quadratic equation in the form \(ax^2 + bx + c = 0\) with \(a = 4\), \(b = -4\), and \(c = -15\).
02
Multiplying to find factor pairs
We need to factor the quadratic expression. The first step is to multiply the coefficient of \(x^2\), which is 4, by the constant term, which is -15. That gives us \(4 imes (-15) = -60\). We are looking for two numbers that multiply to -60 and add to the coefficient of \(x\), which is -4.
03
Finding the pair of numbers
The numbers that multiply to -60 and add to -4 are 6 and -10.
04
Rewrite the middle term
Use the numbers found in Step 3 to rewrite the equation: \(4x^2 - 10x + 6x - 15 = 0\).
05
Factor by grouping
Next, group the terms and factor each group separately: - Group 1: \(4x^2 - 10x\) can be factored as \(2x(2x - 5)\). - Group 2: \(6x - 15\) can be factored as \(3(2x - 5)\). So we have: \(2x(2x - 5) + 3(2x - 5) = 0\).
06
Factor out the common binomial
Now, factor out the common factor, \((2x - 5)\): \((2x - 5)(2x + 3) = 0\).
07
Solve for x
Set each factor equal to zero and solve for \(x\):- \(2x - 5 = 0\) gives \(x = \frac{5}{2}\). - \(2x + 3 = 0\) gives \(x = -\frac{3}{2}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Solving Quadratic Equations
Solving quadratic equations is a fundamental skill in algebra. These equations take the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents an unknown value. Quadratic equations are typically solved using methods such as:
- Factoring
- Using the quadratic formula
- Completing the square
Factoring by Grouping
Factoring by grouping is a method used to factor polynomials by splitting a middle term into two terms, based on specific criteria. This technique is especially useful in quadratic equations like \( 4x^2 - 4x - 15 = 0 \). In this method, you rewrite the quadratic equation in a form that allows for rearranging terms in pairs
- Find two numbers that multiply to \( a \cdot c \) and add to \( b \)
- Rewrite the quadratic with these new terms inserted, creating four terms
- Group terms two-by-two and factor the greatest common factors from each
- Factor out the common binomial factor from the two grouped expressions
Quadratic Formula
The quadratic formula is a powerful tool for solving any quadratic equation. It can always be used, even when other methods like factoring or completing the square are not feasible. The formula is given as:\[x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\]This formula calculates the roots by plugging in the values of \( a \), \( b \), and \( c \) from the standard form of the quadratic equation \( ax^2 + bx + c = 0 \). Using the quadratic formula involves four essential steps:
- Identifying coefficients \( a \), \( b \), and \( c \)
- Calculating the discriminant \( b^2 - 4ac \)
- Applying the values in the quadratic formula
- Simplifying to find the roots
Roots of Equations
The roots of a quadratic equation are the values of \( x \) that make the equation true, meaning they satisfy \( ax^2 + bx + c = 0 \). These roots can be real or complex numbers, depending on the discriminant \( b^2 - 4ac \):
- If the discriminant is positive, there are two distinct real roots.
- If the discriminant is zero, there is exactly one real root (a repeated root).
- If the discriminant is negative, the equation has two complex roots.
