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Chapter 8: Problem 8

Use the quadratic formula to solve each equation. (All solutions for these equations are real numbers.) $$ 8 x^{2}+10 x-3=0 $$

Short Answer

Expert verified
\( \frac{1}{4} \) and \( -\frac{3}{2} \)

Step by step solution

01

Identify coefficients

For the quadratic equation in standard form, identify the coefficients. The standard form is given by: \[ ax^2 + bx + c = 0 \] In this equation, \( a = 8 \), \( b = 10 \), and \( c = -3 \).
02

Write the quadratic formula

The quadratic formula to solve an equation of the form \( ax^2 + bx + c = 0 \) is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
03

Calculate the discriminant

Calculate the discriminant which is the expression under the square root in the quadratic formula: \[ b^2 - 4ac \] Substituting the coefficients: \[ (10)^2 - 4(8)(-3) = 100 + 96 = 196 \]
04

Compute the square root of the discriminant

Find the square root of the discriminant: \[ \sqrt{196} = 14 \]
05

Apply the quadratic formula

Substitute the values of \( b \), \( \sqrt{b^2 - 4ac} \), and \( a \) back into the quadratic formula: \[ x = \frac{-10 \pm 14}{2(8)} \]
06

Calculate the two solutions

Solve for the two possible values of \( x \): \[ x = \frac{-10 + 14}{16} = \frac{4}{16} = \frac{1}{4} \] and \[ x = \frac{-10 - 14}{16} = \frac{-24}{16} = -\frac{3}{2} \]

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Key Concepts

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

Quadratic Equations
Quadratic equations are fundamental in algebra and describe a wide range of phenomena. A quadratic equation is any equation that can be written in the form \[ ax^2 + bx + c = 0 \] where:
  • \(a\) is the coefficient of the squared term \(x^2\)
  • \(b\) is the coefficient of the linear term \(x\)
  • \(c\) is the constant term
These equations are called 'quadratic' because 'quad' means 'square' and they always involve an \( x^2 \) term. The highest power of the variable \(x\) is \(2\). When solving quadratic equations, one efficient method used is the quadratic formula.
Discriminant
The discriminant is a key part of the quadratic formula and helps determine the nature of the solutions. The discriminant is the term under the square root in the quadratic formula: \[ b^2 - 4ac \] Depending on the value of the discriminant, we can determine the types of solutions to our quadratic equation:
  • If \( b^2 - 4ac > 0 \), there are two distinct real solutions.
  • If \( b^2 - 4ac = 0 \), there is one real solution (a repeated root).
  • If \( b^2 - 4ac < 0 \), there are no real solutions; instead, the solutions are complex or imaginary.
In our exercise, the discriminant calculated was \[ 196 \] Since this is greater than zero, it confirms there are two distinct real number solutions.
Real Number Solutions
Real number solutions are the points where the quadratic equation crosses the x-axis on a graph. Solving for these involves using the quadratic formula \[ x = \frac{-b \,\pm \sqrt{b^2 - 4ac}}{2a} \] For the specific example:
  • Our coefficients were \(a = 8\), \( b = 10 \), and \( c = -3 \).
  • The discriminant was determined to be \( 196 \), a positive number, indicating two distinct real solutions.
  • Substituting back into the quadratic formula gave two solutions: \( x = \frac{1}{4} \) and \( x = -\frac{3}{2} \).
These solutions signify the exact points where the parabola described by the quadratic equation intersects the x-axis.

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Most popular questions from this chapter

A model rocket is projected vertically upward from the ground. Its distance s in feet above the ground after t seconds is given by the quadratic function $$ s(t)=-16 t^{2}+256 t $$ to see how quadratic equations and inequalities are related. At what times will the rocket be less than 624 ft above the ground? (Hint: Let \(s(t)<624,\) solve the quadratic inequality, and observe the solutions in to determine the least and greatest possible values of \(t .)\)

Extending Skills Find the value of \(a, b\), or \(c\) so that each equation will have exactly one rational solution. One solution of \(3 x^{2}-7 x+c=0\) is \(\frac{1}{3}\). Find \(c\) and the other solution.

Solve each equation. Check the solutions. $$9 x^{4}-25 x^{2}+16=0$$

The following exercises are not grouped by type. Solve each equation. $$12 x^{4}-11 x^{2}+2=0$$

A model rocket is projected vertically upward from the ground. Its distance s in feet above the ground after t seconds is given by the quadratic function $$ s(t)=-16 t^{2}+256 t $$ to see how quadratic equations and inequalities are related. At what times will the rocket be more than \(624 \mathrm{ft}\) above the ground? (Hint: Let \(s(t)>624\) and solve the quadratic inequality.)
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