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Magazine Chapter 6: Problem 79
$$ \text { For Problems 51-80, solve each equation. (Objective 2) } $$ $$ 18 x^{2}+55 x-28=0 $$
Short Answer
Expert verified
The solutions are \(x = \frac{4}{9}\) and \(x = -\frac{7}{2}\).
Step by step solution
01
Identify the Type of Equation
The given equation is \(18x^2 + 55x - 28 = 0\). It's a quadratic equation in standard form \(ax^2 + bx + c = 0\) where \(a = 18\), \(b = 55\), and \(c = -28\).
02
Use the Quadratic Formula
Recall the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). We will use this formula to find the values of \(x\).
03
Calculate the Discriminant
Calculate the discriminant \(b^2 - 4ac\): \(b^2 = 55^2 = 3025\)\(4ac = 4 \times 18 \times (-28) = -2016\)Thus, the discriminant is \(3025 - (-2016) = 5041\).
04
Compute the Square Root of the Discriminant
The square root of the discriminant is \(\sqrt{5041} = 71\).
05
Apply the Quadratic Formula
Substitute the values into the quadratic formula: \(x = \frac{-55 \pm 71}{36}\).
06
Solve for the Two Possible Values of x
Calculate the two possible values of \(x\):\(x_1 = \frac{-55 + 71}{36} = \frac{16}{36} = \frac{4}{9}\)\(x_2 = \frac{-55 - 71}{36} = \frac{-126}{36} = -\frac{7}{2}\)
07
Present the Solutions
The solutions to the quadratic equation \(18x^2 + 55x - 28 = 0\) are \(x = \frac{4}{9}\) and \(x = -\frac{7}{2}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Formula
The quadratic formula is a powerful tool for solving quadratic equations. A quadratic equation is any equation that can be rearranged into the form \( ax^2 + bx + c = 0 \). The coefficients \( a \), \( b \), and \( c \) represent known numbers, with \( a \) not equal to zero. If \( a \) were zero, the equation would not be quadratic.
The quadratic formula provides a method to find the values of \( x \) (also known as the roots or solutions) that satisfy the equation. It is given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Here's a simple breakdown of how it works:
The quadratic formula provides a method to find the values of \( x \) (also known as the roots or solutions) that satisfy the equation. It is given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Here's a simple breakdown of how it works:
- -b: This part reverses the sign of the \( b \) coefficient.
- \( \sqrt{b^2 - 4ac} \): The expression under the square root is known as the discriminant, which we'll discuss later.
- The \( \pm \) symbol indicates that there are two solutions: one with a plus and one with a minus.
- 2a: This divides the entire expression by two times the coefficient \( a \), splitting the solution process into two parts for the two roots.
Discriminant
The discriminant is a crucial part of the quadratic formula and plays a major role in determining the nature of the roots of a quadratic equation. It is specifically the part of the quadratic formula found under the square root sign, represented by the expression \( b^2 - 4ac \).
Calculating the discriminant helps in predicting the type of roots the quadratic equation will have. Here are the possibilities:
Calculating the discriminant helps in predicting the type of roots the quadratic equation will have. Here are the possibilities:
- If \( b^2 - 4ac > 0 \): There are two distinct real roots. This means the quadratic equation crosses the x-axis at two different points.
- If \( b^2 - 4ac = 0 \): There is exactly one real root, usually referred to as a repeated or double root. The graph of the quadratic equation touches the x-axis at just one point.
- If \( b^2 - 4ac < 0 \): There are no real roots. Instead, the equation has two complex roots, meaning it does not cross the x-axis at all.
Roots of a Quadratic Equation
The roots of a quadratic equation are the solutions for \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). These roots are essentially the x-coordinates where the parabola intersects the x-axis in its graphical form.
Using the quadratic formula, you can find these roots systematically. For the problem given in the exercise:
Using the quadratic formula, you can find these roots systematically. For the problem given in the exercise:
- After calculating the discriminant (5041), it was determined that there are two distinct real roots.
- Applying the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) with specific values of \( a = 18 \), \( b = 55 \), and \( c = -28 \), we computed the two roots:
- \( x_1 = \frac{4}{9} \)
- \( x_2 = -\frac{7}{2} \)
