Problem 132 - A Graphical Approach to Precalculus with Limits

Question

Find the values of \(b\) for which the equation \(x^{2}+b x+4=0\) has two real solutions.

Step-by-Step Solution

Answer

Values of \(b\) are: \(b < -4\) or \(b > 4\).

Step 1: Identify the Discriminant

To determine the number of real solutions for the quadratic equation \(ax^2 + bx + c = 0\), we need to use the discriminant, \(\Delta\). For the equation \(x^2 + bx + 4 = 0\), the coefficients are \(a = 1\), \(b = b\), and \(c = 4\). The discriminant formula is \(\Delta = b^2 - 4ac\).

Step 2: Substitute into the Discriminant Formula

Substitute the values of \(a\), \(b\), and \(c\) into the discriminant formula: \[\Delta = b^2 - 4 \times 1 \times 4 = b^2 - 16.\]

Step 3: Set Condition for Two Real Solutions

For the quadratic equation to have two real solutions, the discriminant must be greater than zero. Therefore, set up the inequality \(b^2 - 16 > 0\).

Step 4: Solve the Inequality

Solve the inequality \(b^2 - 16 > 0\) to find the range of \(b\). Begin by factoring the expression: \[(b - 4)(b + 4) > 0.\] This inequality holds true when \(b < -4\) or \(b > 4\).

Step 5: Conclusion

The values of \(b\) that make the equation \(x^2 + bx + 4 = 0\) have two real solutions are \(b < -4\) or \(b > 4\).

Key Concepts

Discriminant

The discriminant is a powerful tool in determining the nature of solutions for quadratic equations. It is a part of the quadratic formula, represented as \( \Delta = b^2 - 4ac \). The symbol \( \Delta \) is used for the discriminant and helps in easily finding out whether the solutions of a quadratic equation are real or not.
For the quadratic equation \( ax^2 + bx + c = 0 \), the discriminant is calculated using the coefficients \( a \), \( b \), and \( c \). This is because different values of the discriminant tell us about the nature of the roots without actually solving the equation. This makes it an essential tool for quickly understanding the character of the solutions.

• If \( \Delta > 0 \), the equation has two distinct real solutions.
• If \( \Delta = 0 \), the equation has exactly one real solution, which is a repeated root.
• If \( \Delta < 0 \), the equation has no real solutions, and the roots are complex or imaginary.

In our exercise, the discriminant needs to be greater than zero for two real solutions. Therefore, solving \( b^2 - 16 > 0 \) helps identify those cases.

Real Solutions

Real solutions are the values of \( x \) that satisfy the quadratic equation and are real numbers. Whether an equation has real or complex solutions depends on its discriminant. Two real solutions occur when \( \Delta > 0 \).
These solutions are called distinct real roots. They show that the quadratic graph, which is a parabola, crosses the x-axis at two different points. When the discriminant is zero, even though there is technically only one solution because both solutions are the same, we still consider it 'real'
since both solutions overlap at a single x-axis point.

• For our equation \( x^2 + bx + 4 = 0 \), real solutions mean that the value of \( b \) leads to a discriminant greater than zero, allowing the parabola to intersect the x-axis twice.

Inequality

Inequalities are mathematical expressions showing that one quantity is greater than or less than another. When dealing with discriminants to determine the nature of quadratic solutions, inequalities are fundamental. To solve for two real solutions, the inequality \( b^2 - 16 > 0 \) must be satisfied. This determines the values of \( b \) that will ensure the discriminant is positive.
Let's break down the inequality \( b^2 - 16 > 0 \):

• Factoring it gives \((b - 4)(b + 4)> 0\). This indicates that the product is positive when both factors are either positive or both negative.
• The solution \( b < -4 \) or \( b > 4 \) comes from understanding the factors. When \( b < -4 \), both \((b - 4)\) and \((b + 4)\) are negative; when \( b > 4 \), both are positive.

These insights help determine why particular values of \( b \) lead to the equation having two real solutions, providing a deeper understanding of quadratic equations and their solutions.