A quadratic equation ƒ(x) = 0 has a solution x = 2. Its graph has vertex (5, 3). What is the other solution of the equation?

Verified step by step guidance

1. Identify the vertex form of a quadratic equation: \( f(x) = a(x-h)^2 + k \), where \((h, k)\) is the vertex.
2. Substitute the vertex \((5, 3)\) into the vertex form: \( f(x) = a(x-5)^2 + 3 \).
3. Since \( x = 2 \) is a solution, substitute \( x = 2 \) into the equation and set \( f(x) = 0 \): \( 0 = a(2-5)^2 + 3 \).
4. Solve for \( a \) using the equation from the previous step.
5. Use the value of \( a \) to write the complete quadratic equation and find the other solution by setting \( f(x) = 0 \) and solving for \( x \).

Key Concepts

Quadratic Equations

A quadratic equation is a polynomial equation of the form ƒ(x) = ax² + bx + c, where a, b, and c are constants, and a ≠ 0. The solutions to this equation, also known as roots, can be found using various methods such as factoring, completing the square, or the quadratic formula. The graph of a quadratic equation is a parabola, which can open upwards or downwards depending on the sign of 'a'.

Vertex of a Parabola

The vertex of a parabola is the highest or lowest point on its graph, depending on whether it opens downwards or upwards. For a quadratic function in standard form, the vertex can be found using the formula (-b/2a, ƒ(-b/2a)). In this case, the vertex is given as (5, 3), indicating that the parabola opens upwards and the maximum or minimum value occurs at this point.

Symmetry of Parabolas

Parabolas exhibit symmetry about a vertical line known as the axis of symmetry, which passes through the vertex. For a quadratic equation, if one root is known, the other root can be determined using the axis of symmetry. In this case, if one solution is x = 2 and the vertex is at x = 5, the other solution can be found by reflecting x = 2 across the axis of symmetry, leading to the conclusion that the other solution is x = 8.