Chapter 12: Problem 17
Challenge Problems $$3.88\left(x^{2}+7.72\right)=6.34 x(3.99 x-3.81)+7.33$$
Short Answer
Expert verified
First, expand and simplify both sides, then set the quadratic equation to zero and solve for x using appropriate techniques.
Step by step solution
01
Expand the Equations
Expand both sides of the equation by distributing the multiplication. On the left-hand side multiply 3.88 with each term inside the parentheses, and on the right-hand side multiply 6.34x with both terms inside the parentheses.
02
Simplify Both Sides
After distributing the multiplication, combine like terms on both sides to simplify the equation. This involves combining the constant terms and the terms with the variable x.
03
Set Equation to Zero
Subtract all terms from one side of the equation to set the equation to zero. This will aid in solving the quadratic equation.
04
Solve the Quadratic Equation
Use the quadratic formula, factoring, or completing the square to solve for x in the quadratic equation. Make sure to check for both possible solutions.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Expand Equations
When solving complex equations, especially quadratic ones, expanding the equation is a crucial first step. This means distributing any factors across the terms within parentheses. In our exercise, you'll see terms like \(3.88(x^2 + 7.72)\) and \(6.34x(3.99x - 3.81)\).
To expand, multiply each term in the parentheses by the number or expression outside. For example, \(3.88\) gets multiplied with \(x^2\) and \(7.72\), separately. Similarly, on the right, \(6.34x\) multiplies both \(3.99x\) and \-3.81)\. Breaking down these steps further:
To expand, multiply each term in the parentheses by the number or expression outside. For example, \(3.88\) gets multiplied with \(x^2\) and \(7.72\), separately. Similarly, on the right, \(6.34x\) multiplies both \(3.99x\) and \-3.81)\. Breaking down these steps further:
- \(3.88 * x^2 = 3.88x^2 \)
- \(3.88 * 7.72 = 29.9536 \)
- \(6.34x * 3.99x = 25.3074x^2 \)
- \(6.34x * -3.81 = -24.1554x \)
Simplifying Algebraic Expressions
Once you've expanded the equation, the next step is to simplify. This involves combining like terms, which are terms that have the same variable raised to the same power. For instance, in the expression \(3.88x^2 + 29.9536 + 25.3074x^2 - 24.1554x\), there are two types of like terms: those with \(x^2\) and those with \(x\).
To simplify, combine these like terms by adding or subtracting them:
To simplify, combine these like terms by adding or subtracting them:
- \(3.88x^2 + 25.3074x^2 = 29.1874x^2\)
- Since there are no other terms with \(x\), we leave \-24.1554x\ as it is.
Quadratic Formula
If your quadratic equation cannot be factored easily or at all, the quadratic formula is the silver bullet. It's a standardized way to find the roots of any quadratic equation of the form \(ax^2 + bx + c = 0\). The quadratic formula is expressed as \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
This formula applies to all quadratic equations, and provides two solutions, because of the \(\pm\) sign. It accounts for both the possibility of the square root being added to or subtracted from \(\-b\). When using the quadratic formula, accuracy in calculation is vital. You'll need to accurately identify and substitute the coefficients \(a\), \(b\), and \(c\) from the equation. After simplification, your equation will fit the \(ax^2 + bx + c = 0\) form perfectly for application of the quadratic formula.
This formula applies to all quadratic equations, and provides two solutions, because of the \(\pm\) sign. It accounts for both the possibility of the square root being added to or subtracted from \(\-b\). When using the quadratic formula, accuracy in calculation is vital. You'll need to accurately identify and substitute the coefficients \(a\), \(b\), and \(c\) from the equation. After simplification, your equation will fit the \(ax^2 + bx + c = 0\) form perfectly for application of the quadratic formula.
Factoring Quadratic Equations
Factoring is another effective method for solving quadratic equations, particularly when the equation is in a factorable form. Factoring involves rewriting the equation as a product of binomials. Take a quadratic equation \(ax^2 + bx + c = 0\); the factored form might look like \(a(x-n)(x-m) = 0\), where \(n\) and \(m\) are the solutions to the equation.
To factor a quadratic equation, look for two numbers that both add up to \(b\) and multiply to \(ac\). It requires a bit of trial and error, and sometimes factoring by grouping or other advanced tactics if the equation does not factorize simply. Always check for factors that are common to all terms, as simplifying first can make the process smoother.
To factor a quadratic equation, look for two numbers that both add up to \(b\) and multiply to \(ac\). It requires a bit of trial and error, and sometimes factoring by grouping or other advanced tactics if the equation does not factorize simply. Always check for factors that are common to all terms, as simplifying first can make the process smoother.