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?

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1. Recall that the quadratic function can be written in vertex form as \(f(x) = a(x - h)^2 + k\), where \((h, k)\) is the vertex. Here, the vertex is given as \((5, 3)\), so the function can be expressed as \(f(x) = a(x - 5)^2 + 3\).
2. Since \(x = 2\) is a solution to the equation \(f(x) = 0\), substitute \(x = 2\) and \(f(x) = 0\) into the vertex form to find the value of \(a\): \(0 = a(2 - 5)^2 + 3\).
3. Simplify the equation to solve for \(a\): calculate \((2 - 5)^2\) and then isolate \(a\) on one side of the equation.
4. Once \(a\) is found, write the quadratic function explicitly as \(f(x) = a(x - 5)^2 + 3\).
5. To find the other solution, set \(f(x) = 0\) and solve the equation \(a(x - 5)^2 + 3 = 0\) for \(x\). This will give two solutions, one of which is \(x = 2\), and the other will be the other solution you are asked to find.

Key Concepts

Quadratic Equation and Its Roots

A quadratic equation is a second-degree polynomial of the form ax² + bx + c = 0. It has two solutions or roots, which can be real or complex. These roots correspond to the x-values where the graph of the quadratic function intersects the x-axis.

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. Knowing the vertex helps in understanding the shape and position of the parabola, and it can be used to find the quadratic equation when combined with other points.

Symmetry of a Parabola

A parabola is symmetric about a vertical line called the axis of symmetry, which passes through the vertex. If one root is known, the other root can be found by reflecting it across the axis of symmetry, using the vertex's x-coordinate as the midpoint between the roots.