Problem 57 - Precalculus : Building Concepts and Connections

Question

Solve the quadratic equation using any method. Find only real solutions. $$-x^{2}+2 x=1$$

Step-by-Step Solution

Answer

The only real solution for the quadratic equation \(-x^{2}+2 x = 1\) is \(x = 1\).

Step 1: Rearrange the equation

Rearrange the equation \(-x^{2}+2 x = 1\), to get it into the standard form of a quadratic equation \(ax^{2} + bx + c = 0\). This can be obtained as follows: \(-x^{2}+2 x - 1 = 0\). Now, your a, b, and c are -1, 2, and -1 respectively.

Step 2: Substitute into the quadratic formula

The quadratic formula is \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\). Substitute the values a = -1, b = 2, and c = -1 into this formula to solve for x as follows: \(x = \frac{-2 \pm \sqrt{2^{2} - 4*(-1)*(-1)}}{2*-1}\).

Step 3: Compute inside the square root & denominator

Simplify the equation inside the square root and the denominator individually first: \(2^{2} - 4*(-1)*(-1) = 4 - 4 = 0\) for inside the square root and \(2*-1 = -2\) for the denominator. Now, the equation becomes \(x = \frac{-2 \pm \sqrt{0}}{-2}\).

Step 4: Solve for x

With our current equation, \(x = \frac{-2 \pm \sqrt{0}}{-2}\), simplify further to get the values of x. We'll have \(x = \frac{-2 \pm 0}{-2}\), which gives us two possibilities: \(x = \frac{-2 + 0}{-2} = 1\) and \(x = \frac{-2 - 0}{-2} = 1\). Hence, x = 1 is the only (repeated) real solution to the equation.

Key Concepts

The Quadratic Formula

When it comes to solving quadratic equations, the quadratic formula is a reliable and powerful tool that students can use. It states that for any quadratic equation in the standard form, \(ax^2 + bx + c = 0\), the solutions for x can be found as \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Let's break this down:

• a, b, and c are known as the coefficients from the standard form of the equation, with a being the coefficient of \(x^2\), b being the coefficient of x, and c as the constant term.
• The symbol \(\pm\) indicates that there will be two solutions: one with a plus and one with a minus.
• The expression under the square root, \(b^2 - 4ac\), is known as the discriminant and it plays a crucial role in determining the nature and number of the solutions.

Applying the quadratic formula involves just a few steps. First, identify your coefficients a, b, and c from the given equation. Next, plug these values into the formula and simplify. The terms inside the square root and the denominator must be computed before determining the final values for x, as demonstrated in our step by step solution.

Real Solutions of Quadratics

When we talk about real solutions of quadratics, we refer to the values of x that satisfy the quadratic equation and which also can be plotted on the real number line. These solutions might be two distinct real numbers, one real repeated solution, or none at all--in the case of complex solutions.

The discriminant of the quadratic equation, which is the part under the square root in the quadratic formula (\(b^2 - 4ac\)), determines the number and type of solutions:

• Positive Discriminant: If \(b^2 - 4ac > 0\), the equation has two distinct real solutions.
• Zero Discriminant: If \(b^2 - 4ac = 0\), the equation has exactly one real solution, also known as a repeated or double root.
• Negative Discriminant: If \(b^2 - 4ac < 0\), there are no real solutions; instead, there are two complex solutions.

Therefore, the discriminant not only provides information about the existence of real solutions but also how many there are. In our original exercise, because the discriminant was 0, we concluded there is one real repeated solution for x.

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is an essential format that allows us to analyze and solve these equations systematically. It is represented as \(ax^2 + bx + c = 0\), where a, b, and c are constants, and a is not zero. To solve a quadratic equation, one of the first steps is often to rewrite it in this standard form, as seen in our exercise.

Here's why the standard form is quite useful:

• It allows immediate identification of the coefficients needed for the quadratic formula.
• It makes it easier to graph the equation by identifying the vertex and direction of the parabola.
• It sets a uniform basis for factoring or completing the square, other methods of solving quadratics.

In the provided exercise, rearranging to the standard form made it possible to identify the values \(a = -1\), \(b = 2\), and \(c = -1\). This preliminary step is crucial, as using incorrect coefficients may lead to the wrong roots of the equation. Once in standard form, we applied the quadratic formula to find the real solution.