Question

Explain how to use the quadratic formula to solve \(-2 x^{2}+5 x=-7\).

Short answer

The solutions to the equation -2x^2 + 5x + 7 = 0 are \( x = -1 \) and \( x = 3.5 \)

Step-by-step solution

Rewrite the equation in standard form

The given equation is \( -2x^2 + 5x = -7 \). Let's add 7 to both sides of the equation so it matches the standard form \( ax^2 + bx + c = 0 \). This gives us \( -2x^2 + 5x + 7 = 0 \). So, \( a = -2 \), \( b = 5 \), and \( c = 7 \)

Substitute the values in the quadratic formula

After obtaining the values of a, b, and c, we substitute them into our quadratic formula. Doing this yields:\( x = \frac{-b \pm \sqrt {b^2 - 4ac}}{2a} \) becomes \( x = \frac{-5 \pm \sqrt {(5)^2 - 4(-2)(7)}}{2(-2)} \)

Simplify the equation

First, calculate the values inside the square root and the denominator. For the square root operation, we get \( 5^2 - 4*(-2)*7 = 25 + 56 = 81 \). The denominator is \( 2 * -2 = -4 \). Substituting these values back into the equation, we get \( x = \frac{-5 \pm \sqrt {81}}{-4} \)

Calculate the solutions

Now calculate the two roots of the equation. The \pm indicates that there are two solutions: one using a plus sign and the other using a minus sign. This gives us \( x_1 = \frac{-5 + \sqrt {81}}{-4} \) and \( x_2 = \frac{-5 - \sqrt {81}}{-4} \). This simplifies to \( x_1 = \frac{-5 + 9}{-4} = -1 \) and \( x_2 = \frac{-5 - 9}{-4} = 3.5 \)

Key concepts

Solving Quadratic Equations

Quadratic equations are an indispensable part of algebra, presenting a level of complexity that is both challenging and rewarding. To solve a quadratic equation means to find the values of the variable (usually denoted as 'x') that satisfy the given equation. The quadratic formula is a widely used method for solving these types of equations. It is a straightforward tool that, when used properly, provides the roots directly and can be easily remembered as: \[\begin{equation} x = \frac{-b \pm \sqrt {b^2 - 4ac}}{2a}\text{, where } a, b, \text{ and } c \text{ are coefficients of the quadratic equation } ax^2 + bx + c = 0.\text{\end{equation}\]}After identifying the coefficients, they are substituted into the formula. This method can handle all types of quadratic equations, including those with real and complex roots. With practice, students can streamline their problem-solving process by quickly recognizing the standard form and efficiently calculating the discriminant—the part under the square root—which determines the nature and number of solutions.

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is expressed as \[\begin{equation}ax^2 + bx + c = 0\text{, where \(a\), \(b\), and \(c\) are constants and \(a eq 0\)}.\text{\end{equation}\]}This form is pivotal for solving quadratic equations by various methods, including factoring, completing the square, graphing, or using the quadratic formula. To utilize these methods effectively, one must often manipulate the original equation to achieve the standard form. This process might include combining like terms, moving all terms to one side of the equation, and ensuring that the leading coefficient of the quadratic term is non-zero. Understanding this standard form also unlocks insights into the properties of a quadratic equation, such as the axis of symmetry of its graph and its vertex, paving the way for a deeper comprehension of quadratic functions.

Roots of a Quadratic Equation

In the context of quadratic equations, the roots—also known as solutions or x-intercepts—are the values of 'x' that satisfy the equation. To envision this geometrically, these roots represent the points where the parabola (the graph of the quadratic equation) crosses the x-axis.Finding the roots is central to many areas of mathematics and applications in physics, engineering, and economics. The quadratic formula is particularly useful in finding these roots, especially when other methods fall short or become too complex to apply. Once the roots are calculated, they can reveal whether they are real or complex, and whether the parabola touches the x-axis at one point (a repeated root) or crosses at two distinct points. It also offers insight into the factors of the quadratic. In simple terms, these roots are the answers to the question, 'For which values of 'x' does this equation hold true?' and embody one of the key goals in solving quadratic equations.