Problem 91 - Algebra 1: Concepts and Skills

Question

Use a graph to estimate the solutions of the equation. Check your solutions algebraically. (Lesson 9.5) $$ 2 x^{2}-3 x+4=0 $$

Step-by-Step Solution

Answer

The exact solutions obtained by solving the equation algebraically may not exactly match the estimated solutions from the graph because of the inaccuracies of estimation. However, they should be quite close.

Step 1: Plot the function

First, create a graph of the given quadratic function, \(2x^2 - 3x + 4\). You need to look at where it cuts the x-axis because these points give the solutions to the equation.

Step 2: Estimate the solutions

Estimate where the graph of the function crosses the x-axis. These x-values are the solutions to the equation. Keep in mind these are just estimates.

Step 3: Check the solutions algebraically

Now check your estimated solutions algebraically. Use the quadratic formula, which is \(-b \pm \sqrt{b^2 - 4ac} \, / \, 2a\), where \(a\), \(b\) and \(c\) are coefficients of the quadratic equation \(ax^2 + bx + c\). Here, a = 2, b = -3 and c = 4. Substitute these values into the formula and you will get the exact solutions.

Key Concepts

Graphing Quadratic Functions

Graphing a quadratic function, like our example \(2x^2 - 3x + 4\), helps us visually analyze the equation. A quadratic function can be represented as a parabola on a graph. Depending on the sign of \(a\) in the standard form \(ax^2 + bx + c\), the parabola either opens upwards (when \(a > 0\)) or downwards (when \(a < 0\)). In this case, since \(a = 2\), the parabola opens upwards.
To accurately draw the parabola, it's important to identify its vertex and axis of symmetry. The vertex form of a quadratic is \(y = a(x-h)^2 + k\), where \((h, k)\) is the vertex. For basic plotting, you can also find intercepts and use symmetrical points for a rough sketch.

• Calculate the vertex using \(x = -\frac{b}{2a}\) for the x-coordinate.
• Substitute this \(x\) into the function for the y-coordinate.

Ultimately, graphing the equation allows us to approximate the solutions by observing where the parabola intersects the x-axis.

Estimating Solutions

Estimating the solutions of a quadratic equation through graphing involves looking for the points where the graph crosses the x-axis, known as the "roots" or "zeroes" of the function. In practice, for our function \(2x^2 - 3x + 4\), if the parabola crosses at or near the x-axis, these x-values would represent potential solutions.
However, if the parabola does not cut through the x-axis, it signifies that there are no real solutions. This occurs when the parabola lies entirely above or below the axis, indicating that the solutions may be complex, rather than real.
Graphing software or graphing calculators can assist in making these estimations with greater accuracy. Always remember estimations are approximate and should be verified with algebraic solutions for precision.

Quadratic Formula

The quadratic formula is a mathematical tool used to find the exact solutions to any quadratic equation of the form \(ax^2 + bx + c = 0\). It's represented as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]In this formula:

• \(a\) is the coefficient of \(x^2\),
• \(b\) is the coefficient of \(x\), and
• \(c\) is the constant term.

Using this method, inputting \(a = 2\), \(b = -3\), and \(c = 4\) into the formula helps determine the precise solutions or roots of the quadratic equation.
When the value inside the square root, called the discriminant \(b^2 - 4ac\), is negative, there are no real roots, only complex ones. This supports our findings in estimating solutions, confirming observation through calculation.

Algebraic Solutions

Algebraically solving a quadratic equation solidifies the approximate results you'd initially estimate graphically. With algebraic solutions, you have the ability to verify and precisely determine the exact roots of the equation.
For our given equation \(2x^2 - 3x + 4\), substituting into the quadratic formula will offer an exact answer.

• The calculation involves simplifying \(-3 \pm \sqrt{(-3)^2 - 4(2)(4)} \over 4\).
• If the discriminant computes to a negative number, solutions will involve imaginary numbers, seen with \(i\), the symbol for \(\sqrt{-1}\).

These solutions are essential in fully understanding the nature of the equation's roots. Numbers computed algebraically match, or clarify, estimations we made by observing the graph's intersections with the x-axis.