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Chapter 8: Problem 3

Find the real solutions of \(4 x^{2}-x-5=0\).

Short Answer

Expert verified
The real solutions are \( x = \frac{5}{4} \) and \( x = -1 \).

Step by step solution

01

Identify the coefficients

The given quadratic equation is \(4x^2 - x - 5 = 0\). Identify the coefficients of the equation, where \(a = 4\), \(b = -1\), and \(c = -5\).
02

Use the quadratic formula

The quadratic formula to find the roots of the equation \(ax^2 + bx + c = 0\) is given by \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. Substitute \(a = 4\), \(b = -1\), and \(c = -5\) into this formula.
03

Calculate the discriminant

First calculate the discriminant \[ \Delta = b^2 - 4ac \]. Substitute the coefficients to get: \( \Delta = (-1)^2 - 4(4)(-5) = 1 + 80 = 81 \).
04

Find the square root of the discriminant

Take the square root of the discriminant value: \( \sqrt{81} = 9 \).
05

Substitute back into the quadratic formula

Substitute the values of \(b\), \(\sqrt{\Delta}\), and \(a\) back into the quadratic formula: \[ x = \frac{-(-1) \pm 9}{2 \cdot 4} \]. This simplifies to: \[ x = \frac{1 \pm 9}{8} \].
06

Solve for the two roots

Calculate the two possible values for \(x\): \[ x_1 = \frac{1 + 9}{8} = \frac{10}{8} = \frac{5}{4} \] and \[ x_2 = \frac{1 - 9}{8} = \frac{-8}{8} = -1 \].

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Key Concepts

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

quadratic formula
To solve any quadratic equation, the quadratic formula is a powerful tool. The formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here, the quadratic formula helps us find the values of \(x\) that satisfy the quadratic equation \(ax^2 + bx + c = 0\). The terms involved in the quadratic formula are:
  • \(a\): Coefficient of \(x^2\)
  • \(b\): Coefficient of \(x\)
  • \(c\): Constant term
In the original problem, the values are \(a = 4\), \(b = -1\), and \(c = -5\). The formula incorporates these coefficients to ultimately find the roots of the equation.
discriminant
The discriminant is a vital part of the quadratic formula and provides important insights into the nature of the roots. It is computed as:\[ \Delta = b^2 - 4ac \]In our example, this becomes:\[ \Delta = (-1)^2 - 4(4)(-5) = 1 + 80 = 81 \]The value of the discriminant (\(\Delta = 81\)) tells us the nature of the roots:
  • If \(\Delta > 0\), there are two distinct real roots.
  • If \(\Delta = 0\), there is exactly one real root (a repeated root).
  • If \(\Delta < 0\), the roots are complex or imaginary numbers.
In our case, the positive discriminant (\(81\)) indicates that the equation \(4x^2 - x - 5 = 0\) has two distinct real roots.
solving quadratic equations
Once we have the coefficients and the discriminant, solving the quadratic equation involves several clear steps:
  • Step 1: Identify the coefficients \(a\), \(b\), and \(c\).
  • Step 2: Calculate the discriminant \(\Delta = b^2 - 4ac\).
  • Step 3: Use the quadratic formula \(x = \frac{-b \pm \sqrt{\Delta}}{2a}\).
  • Step 4: Substitute the coefficients and the calculated discriminant back into the formula.
    • In this case: \(x = \frac{-(-1) \pm \sqrt{81}}{2 \cdot 4}\) simplifies to \(x = \frac{1 \pm 9}{8}\).
  • Step 5: Solve for the two values of \(x\):
    • \(x_1 = \frac{1 + 9}{8} = \frac{10}{8} = \frac{5}{4}\)
    • \(x_2 = \frac{1 - 9}{8} = \frac{-8}{8} = -1\)
With these steps, we find that the solutions to the equation \(4x^2 - x - 5 = 0\) are \(x_1 = \frac{5}{4}\) and \(x_2 = -1\).

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Most popular questions from this chapter

Find the exact value of each expression. $$ \sin \left(\sin ^{-1} \frac{1}{2}+\cos ^{-1} 0\right) $$

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