Which graph shows solving $2 x^2-8 x+5=-x^2+4$ by graphing a system?

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02-11-2026
Mathematics

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Which graph shows solving $2 x^2-8 x+5=-x^2+4$ by graphing a system?

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-02T21:21:44-05:00

Represent the equation as a system of two functions: $y_1 = 2x^2 - 8x + 5$ and $y_2 = -x^2 + 4$.
Analyze the parabolas: $y_1$ is upward-opening with vertex $(2, -3)$; $y_2$ is downward-opening with vertex $(0, 4)$.
Calculate intersection points by setting $2x^2 - 8x + 5 = -x^2 + 4$, yielding $x = \frac{4 \pm \sqrt{13}}{3}$.
Identify the correct graph as the one displaying these specific vertices and intersection points at $x \approx 0.132$ and $x \approx 2.535$.
Final answer: $\boxed{x = \frac{4 \pm \sqrt{13}}{3}}$

Explanation

Analyzing the System of Equations

To solve the equation $2x^2 - 8x + 5 = -x^2 + 4$ by graphing, we interpret the equation as a system of two separate functions, where each side of the equation is set equal to $y$:

$y_1 = 2x^2 - 8x + 5$
$y_2 = -x^2 + 4$

The solutions to the original equation are the x-coordinates of the points where these two parabolas intersect. We will analyze the properties of each parabola to understand how they should appear on a graph.

Determining Key Properties of the Parabolas

For the first function, $y_1 = 2x^2 - 8x + 5$:

The parabola opens upward since the coefficient of $x^2$ is positive ($2 > 0$).
The vertex is at $x = -\frac{b}{2a} = -\frac{-8}{2(2)} = 2$.
Substituting $x = 2$ into the equation gives $y_1 = 2(2)^2 - 8(2) + 5 = 8 - 16 + 5 = -3$. So, the vertex is at $(2, -3)$.
The y-intercept is found by setting $x = 0$, which gives $y_1 = 5$.

For the second function, $y_2 = -x^2 + 4$:

The parabola opens downward since the coefficient of $x^2$ is negative ($-1 < 0$).
The vertex is at $x = 0$, which gives $y_2 = 4$. So, the vertex is at $(0, 4)$.
The x-intercepts are found by setting $y_2 = 0$, giving $x^2 = 4$, so $x = \pm 2$.

Calculating the Intersection Points

To find the exact points of intersection, we set $y_1 = y_2$:
$2x^2 - 8x + 5 = -x^2 + 4$
Rearranging terms to one side gives:
$3x^2 - 8x + 1 = 0$
Using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$:
$x = \frac{8 \pm \sqrt{(-8)^2 - 4(3)(1)}}{2(3)} = \frac{8 \pm \sqrt{64 - 12}}{6} = \frac{8 \pm \sqrt{52}}{6} = \frac{8 \pm 2\sqrt{13}}{6} = \frac{4 \pm \sqrt{13}}{3}$
The intersection x-coordinates are approximately $x \approx 2.535$ and $x \approx 0.132$. A correct graph must show the parabolas $y_1$ and $y_2$ intersecting at these two values of $x$.

Conclusion

The correct graph is the one that shows:

A parabola opening upward with vertex at $(2, -3)$ and y-intercept $(0, 5)$.
A parabola opening downward with vertex at $(0, 4)$ and x-intercepts $(-2, 0)$ and $(2, 0)$.
The intersection points occur at approximately $x \approx 0.132$ and $x \approx 2.535$.

These features confirm the visual representation needed to solve the equation $2x^2 - 8x + 5 = -x^2 + 4$ by graphing.

Examples

Graphing systems of equations is essential in fields like engineering and economics. For instance, if a company wants to find the 'break-even point' where their production cost (modeled by one function) equals their revenue (modeled by another), they can graph both functions and identify the intersection point. By visualizing these relationships, planners can quickly determine the range of production where the business is profitable, making complex algebraic comparisons intuitive and accessible.