Question

USING EQUATIONS How many solutions does the system have? Explain your reasoning. $$ \begin{aligned} &y=7 x^2-11 x+9 \\ &y=-7 x^2+5 x-3 \end{aligned} $$ (A) 0 (B) 1 (C) 2 (D) 4

Short answer

(C) 2

Step-by-step solution

Equate the two functions

Start by setting the two equations equal to each other to find the values of \(x\) where the equations intersect: \(7x^2 - 11x + 9 = -7x^2 + 5x - 3\)

Simplify the equation

Combine the similar terms in the left and right side of the equation, which gives \(14x^2 -16x +12 = 0\)

Solve for the roots

The roots of the equation can be found by either factoring, completing the square or using the quadratic formula. Here, the quadratic formula \(x = [-b ± sqrt(b^2 - 4ac)] / (2a)\) will be used. Where \(a = 14, b = -16, c = 12\). By substituting these values in, we obtain \(x = 3/7 , 4/7\). Therefore, there are two solutions for the system of equations

Key concepts

Quadratic Equations

Quadratic equations are a type of polynomial equation, characterized by having an exponent of 2 for at least one of its terms. The general form is written as \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are real numbers, and \( a eq 0 \). The quadratic equation is fundamental in algebra because it allows us to model various real-world scenarios involving area, projectile motion, and more. In practical applications, solving quadratic equations means finding the value(s) of \( x \) that make the equation true. These values are known as the roots of the equation.

Quadratic equations can have:

• Two distinct solutions
• One repeated solution
• No real solutions

The nature of the solutions depends on the discriminant, \( b^2 - 4ac \), which is part of the quadratic formula. If this value is positive, the equation has two distinct real roots. If it's zero, there's exactly one repeated real root. If negative, the roots are complex numbers.

Intersecting Solutions

When dealing with systems of equations, finding intersecting solutions means identifying the points at which the graphs of the equations meet. For a system involving quadratic equations, these intersections illustrate when and where two parabolic curves cross each other on a graph.

In the context of the provided exercise, the system of equations is given by two quadratic functions. To determine their intersecting solutions, you set the equations equal to each other: \( 7x^2 - 11x + 9 = -7x^2 + 5x - 3 \). This setup helps us find the x-values where both equations yield the same y-value.

Solving this results in a new quadratic equation that gives the x-coordinates of the intersection points. By substituting back into either original equation, the corresponding y-values are found, marking the precise intersection points on the graph.

Quadratic Formula

The quadratic formula is a universal tool to solve any quadratic equation of the form \( ax^2 + bx + c = 0 \). It provides a way to find the roots, even when they cannot be easily factored. The formula is written as:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This formula works based on the discriminant \( b^2 - 4ac \), which helps determine the nature of the roots. The \( \pm \) symbol indicates that there can be two possible solutions, reflecting the parabolic shape's typical nature.

In our exercise, applying the quadratic formula to the equation \( 14x^2 -16x +12 = 0 \) after simplification results in two solutions:\( x = \frac{3}{7} \) and \( x = \frac{4}{7} \). These solutions represent the x-values where the original quadratic equations intersect.

Algebraic Simplification

Algebraic simplification is the process of making an equation more manageable by combining like terms and reducing the equation into a simpler form without changing its solutions. This is crucial when solving systems of equations, especially those involving quadratics.

In our exercise, we start by setting the two quadratic equations equal to each other.\( 7x^2 - 11x + 9 = -7x^2 + 5x - 3 \). By moving terms around and simplifying, we combine like terms, yielding the simplified equation \( 14x^2 -16x +12 = 0 \). Through simplification, complex problems become easier to handle and solve, enabling us to apply methods like the quadratic formula more effectively. This step is particularly essential because it reduces potential errors and helps clearly identify the roots of the equation.