Problem 12
Question
Use the quadratic formula to solve each of the quadratic equations. Check your solutions by using the sum and product relationships. $$x^{2}+4 x-1=0$$
Step-by-Step Solution
Verified Answer
The solutions are \(-2 + \sqrt{5}\) and \(-2 - \sqrt{5}\).
1Step 1: Identify the coefficients
The given equation is \(x^2 + 4x - 1 = 0\). To use the quadratic formula, first identify the coefficients from the equation: \(a = 1\), \(b = 4\), and \(c = -1\).
2Step 2: Write down the quadratic formula
The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This formula will be used to find the roots of the quadratic equation.
3Step 3: Calculate the discriminant
The discriminant is \(b^2 - 4ac\). Substitute \(b = 4\), \(a = 1\), and \(c = -1\) into the formula: \(4^2 - 4 \times 1 \times (-1) = 16 + 4 = 20\). The discriminant is 20.
4Step 4: Substitute into the quadratic formula
Now substitute \(b = 4\), \(a = 1\), and the discriminant value into the quadratic formula: \(x = \frac{-4 \pm \sqrt{20}}{2 \times 1} = \frac{-4 \pm \sqrt{20}}{2}\).
5Step 5: Simplify the expression
Simplify \(\sqrt{20}\) which equals \(2\sqrt{5}\). So the expression becomes \(x = \frac{-4 \pm 2\sqrt{5}}{2}\). Simplify further: \(x = -2 \pm \sqrt{5}\).
6Step 6: Verify using sum and product
The sum of the roots \(x_1 + x_2\) should equal \(-b/a = -4/1 = -4\) and the product \(x_1 \times x_2\) should equal \(c/a = -1/1 = -1\). The roots found are \(-2 + \sqrt{5}\) and \(-2 - \sqrt{5}\). Verify: \((-2 + \sqrt{5}) + (-2 - \sqrt{5}) = -4\) and \((-2 + \sqrt{5})(-2 - \sqrt{5}) = (-2)^2 - (\sqrt{5})^2 = 4 - 5 = -1\), both conditions are satisfied.
Key Concepts
DiscriminantSum and Product of RootsQuadratic Equations
Discriminant
When tackling a quadratic equation using the quadratic formula, one essential piece of the puzzle is the **discriminant**. The discriminant determines the nature and the number of roots a quadratic equation can have.
The discriminant is a part of the quadratic formula given by the expression \(b^2 - 4ac\). Here, \(a\), \(b\), and \(c\) are coefficients from the standard quadratic equation form \(ax^2 + bx + c = 0\).
In the example problem of \(x^2 + 4x - 1 = 0\), the discriminant calculated is 20. Since 20 is a positive number, this equation has two distinct real roots.
The discriminant is a part of the quadratic formula given by the expression \(b^2 - 4ac\). Here, \(a\), \(b\), and \(c\) are coefficients from the standard quadratic equation form \(ax^2 + bx + c = 0\).
- If the discriminant is positive, the quadratic equation has two distinct real roots.
- If the discriminant is zero, the equation has exactly one real root, meaning it's a perfect square trinomial.
- When the discriminant is negative, the roots are complex and not real numbers.
In the example problem of \(x^2 + 4x - 1 = 0\), the discriminant calculated is 20. Since 20 is a positive number, this equation has two distinct real roots.
Sum and Product of Roots
The **sum and product of roots** are insightful properties of quadratic equations. These relationships come in handy as a quick check to verify calculated roots.
For a quadratic equation \(ax^2 + bx + c = 0\), the
In practice, for the equation \(x^2 + 4x - 1 = 0\), with \(a = 1\), \(b = 4\), and \(c = -1\):
So, when the roots \(-2 + \sqrt{5}\) and \(-2 - \sqrt{5}\) are calculated, their sum and product can be checked against these formulas to verify the solution’s accuracy.
For a quadratic equation \(ax^2 + bx + c = 0\), the
- Sum of the roots, denoted \(x_1 + x_2\), is always equal to \(-b/a\).
- The product of the roots, denoted \(x_1\times x_2\), will be \(c/a\).
In practice, for the equation \(x^2 + 4x - 1 = 0\), with \(a = 1\), \(b = 4\), and \(c = -1\):
- The sum of the roots should equal \(-4/1 = -4\).
- The product of the roots should equal \(-1/1 = -1\).
So, when the roots \(-2 + \sqrt{5}\) and \(-2 - \sqrt{5}\) are calculated, their sum and product can be checked against these formulas to verify the solution’s accuracy.
Quadratic Equations
**Quadratic equations** are a form of polynomial equations that are categorized by the highest power of the variable being two. They usually take the standard form \(ax^2 + bx + c = 0\). In this equation:
The quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) allows for the precise calculation of the roots of any quadratic equation, where \(b^2 - 4ac\) is the discriminant.
Understanding these characteristics aids in effectively solving quadratic equations and comprehending their graphical interpretation as parabolas. The roots represent the points at which the parabola intersects the x-axis, providing valuable insights about the solutions to the equation in real-world contexts.
- \(a\) is the coefficient of the quadratic term \(x^2\).
- \(b\) is the coefficient of the linear term \(x\).
- \(c\) is the constant term.
The quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) allows for the precise calculation of the roots of any quadratic equation, where \(b^2 - 4ac\) is the discriminant.
Understanding these characteristics aids in effectively solving quadratic equations and comprehending their graphical interpretation as parabolas. The roots represent the points at which the parabola intersects the x-axis, providing valuable insights about the solutions to the equation in real-world contexts.
Other exercises in this chapter
Problem 12
Solve each inequality and graph its solution set on a number line. $$x(x+3)(x-3) \leq 0$$
View solution Problem 12
Solve each quadratic equation using the method that seems most appropriate to you. $$(x+3)(2 x+1)=-3$$
View solution Problem 12
Solve each quadratic equation by using (a) the factoring method and (b) the method of completing the square. $$2 n^{2}+7 n-4=0$$
View solution Problem 12
Solve each of the quadratic equations by factoring and applying the property, \(a b=0\) if and only if \(a=0\) or \(b=0\). If necessary, return to Chapter 3 and
View solution