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Magazine Chapter 6: Problem 30
Solve. $$ x_{2}-14 x+40=0 $$
Short Answer
Expert verified
The solutions are \( x = 10 \) and \( x = 4 \).
Step by step solution
01
Identify Coefficients
The quadratic equation is given in the form \[ ax^2 + bx + c = 0 \]In the given equation \[ x^2 - 14x + 40 = 0 \], the coefficients are: - \( a = 1 \)- \( b = -14 \)- \( c = 40 \).
02
Apply the Quadratic Formula
The quadratic formula is given by \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].Substitute the values of \( a \), \( b \), and \( c \) from the equation into the formula: \[ x = \frac{-(-14) \pm \sqrt{(-14)^2 - 4 \times 1 \times 40}}{2 \times 1} \].
03
Calculate the Discriminant
Calculate the discriminant \( \Delta \) using the formula \[ \Delta = b^2 - 4ac \].Substitute the values:\[ \Delta = (-14)^2 - 4 \times 1 \times 40 = 196 - 160 = 36 \].
04
Solve for Roots
Substitute the discriminant back into the formula to find the roots:\[ x = \frac{14 \pm \sqrt{36}}{2} \].Calculate both the positive and negative potential roots:1. \[ x = \frac{14 + 6}{2} = \frac{20}{2} = 10 \].2. \[ x = \frac{14 - 6}{2} = \frac{8}{2} = 4 \].
05
State the Solutions
The solutions to the quadratic equation are \( x = 10 \) and \( x = 4 \). These are the points where the parabola crosses the x-axis.
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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 of the form \( ax^2 + bx + c = 0 \). It's particularly useful when you can't easily factor the equation. The formula itself is:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula provides the "roots" or solutions to the quadratic equation. By plugging in the values for \( a \), \( b \), and \( c \), you can find where the function crosses the x-axis.
- The expression \( -b \pm \sqrt{b^2 - 4ac} \) is important because it considers both the possibility of a positive and a negative solution, hence the "\( \pm \)".
- These solutions are helpful in many fields, from physics to finance, whenever a second-degree polynomial needs solving.
Discriminant
The discriminant is a specific part of the quadratic formula, represented by \( b^2 - 4ac \). It reveals crucial information about the nature of the roots without calculating them completely.
- If the discriminant is positive, like 36 in our example, two distinct real roots exist. This tells you that the parabola intersects the x-axis at two points.
- If it's zero, there's exactly one real root. This means the vertex of the parabola just touches the x-axis, resulting in one point of intersection.
- If it's negative, the roots are complex or imaginary, occurring when the parabola doesn't intersect the x-axis at all.
Coefficients
Coefficients in a quadratic equation are the numbers in front of the variables \( x^2 \), \( x \), and the constant term. In the equation \( ax^2 + bx + c = 0 \), \( a \), \( b \), and \( c \) are the coefficients:
- \( a \): The coefficient of \( x^2 \) determines the parabola's direction鈥攗pward if positive and downward if negative.
- \( b \): The coefficient of \( x \) affects the parabola's axis of symmetry and its position relative to the y-axis.
- \( c \): The constant term provides the y-intercept, where the graph crosses the y-axis.
