Problem 27 - Basic Technical Mathematics with Calculus

Question

Solve the given quadratic equations using the quadratic formula. If there are no real roots, state this as the answer. Exercises \(3-6\) are the same as Exercises \(13-16\) of Section 7.2. $$x^{2}-0.20 x-0.40=0$$

Step-by-Step Solution

Answer

The real roots are 0.74 and -0.54.

Step 1: Identify the coefficients

For the quadratic equation \(ax^2 + bx + c = 0\), identify the coefficients \(a\), \(b\), and \(c\). Here, \(a = 1\), \(b = -0.20\), and \(c = -0.40\).

Step 2: Write down the quadratic formula

The quadratic formula is given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). We will use this formula to find the roots of the given equation.

Step 3: Calculate the discriminant

The discriminant of the quadratic equation is \(b^2 - 4ac\). Substituting the values, we have \((-0.20)^2 - 4(1)(-0.40)\). Calculate this expression.

Step 4: Simplify the discriminant

Compute the value of the discriminant: \(0.04 + 1.60 = 1.64\). Since the discriminant is positive, there are two real roots.

Step 5: Substitute into the quadratic formula

Substitute the values and discriminant into the quadratic formula: \(x = \frac{-(-0.20) \pm \sqrt{1.64}}{2 \, \cdot \, 1}\). Simplify to find \(x = \frac{0.20 \pm \sqrt{1.64}}{2}\).

Step 6: Calculate the roots

Calculate \(\sqrt{1.64}\) which is 1.28. So the roots are \(x = \frac{0.20 + 1.28}{2}\) and \(x = \frac{0.20 - 1.28}{2}\).

Step 7: Simplify the results

Simplify the expressions to find the roots: \(x = \frac{1.48}{2} = 0.74\) and \(x = \frac{-1.08}{2} = -0.54\). Thus, the roots of the equation are \(x = 0.74\) and \(x = -0.54\).

Key Concepts

Understanding the Discriminant

The discriminant is a crucial part of solving quadratic equations using the quadratic formula. It is represented by the expression under the square root in the formula: \(b^2 - 4ac\). The discriminant helps in determining the nature of the roots of a quadratic equation.

• If the discriminant is positive (\(b^2 - 4ac > 0\)), the quadratic equation has two distinct real roots.
• If it is zero (\(b^2 - 4ac = 0\)), the equation has exactly one real root, also known as a repeated root since both roots are the same.
• If the discriminant is negative (\(b^2 - 4ac < 0\)), the equation has no real roots; instead, the roots are complex numbers.

Understanding the discriminant gives us insight into whether we should expect real solutions or not, without having to solve the equation fully. This makes it a powerful tool for quickly assessing what kind of solutions to anticipate.

The Quadratic Equation Unveiled

A quadratic equation is any polynomial equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. In these equations, \(a\) cannot be zero because if it were, the equation would not remain quadratic (instead, it becomes linear).
The graph of a quadratic equation is a parabola. The shape and direction of the parabola are determined by the coefficient \(a\). If \(a > 0\), the parabola opens upwards, but if \(a < 0\), it opens downwards.

• The solutions or "roots" of the equation are the points where the parabola intersects the x-axis.
• These solutions can be obtained algebraically using various methods: factoring, completing the square, and, most broadly applicable, the quadratic formula.

Understanding quadratic equations helps us solve a wide range of practical problems involving area, projectile motion, and other physical phenomena. It also helps us understand how changes in parameters affect graphical outputs, enhancing our analytical skills.

Exploring Real Roots

Real roots of a quadratic equation are the solutions or values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). These are essentially the x-values where the parabola intersects the x-axis.

• When solving using the quadratic formula, if the discriminant is positive, the equation will have two different real roots. This means the parabola intersects the x-axis at two distinct points.
• If the discriminant is zero, there is exactly one real root, signifying the parabola just touches the x-axis at one point, often referred to as the vertex of the parabola.

Real roots indicate actual solutions that can be observed on a graph or in real-world scenarios. Anytime we calculate real roots, we are determining concrete outcomes for the scenario modeled by the quadratic equation. This direct connection to real-world applications makes understanding real roots vital for both students and practitioners in scientific and engineering fields.