Question

Use a calculator to solve the equation or write no solution. Round the results to the nearest hundredth. $$4 x^{2}-3=57$$

Short answer

The solutions for the equation are approximately \(x = 3.87\) and \(x = -3.87\)

Step-by-step solution

Rearrange the Equation

Isolate the term \(x^{2}\) in the given equation \(4x^{2}-3=57\). Move -3 to the other side of the equation by adding 3 on both sides. The equation would look like this \(4x^{2}=60\)

Solve for \(x^{2}\)

In order to isolate the \(x\) we need to find \(x^{2}\) first. That requires us to divide both sides of the equation by 4. This results in \(x^{2}=15\)

Solve for \(x\)

To solve for \(x\), take the square root of both sides. The result is also should be a positive or negative value. So, \(x = \pm \sqrt{15}\)

Round off to the Nearest Hundredth

Using a calculator, calculate the values of \(\sqrt{15}\) and round to the nearest hundredth. Having performed these calculations, \(x\) takes the values \(\pm 3.87\)

Key concepts

Quadratic Formula

Solving quadratic equations can sometimes be done by rearranging the terms and isolating variables, but when the equation doesn't lend itself to simple methods such as factoring, the quadratic formula comes to the rescue. The quadratic formula is \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \) where \( ax^2 + bx + c = 0 \) is the standard form of a quadratic equation. In the given exercise, if the equation had a middle term (\(bx\)) and wasn't easily factorable, we would apply the quadratic formula. This powerful tool allows us to find the roots of any quadratic equation by simply plugging values into the formula. It's worth noting that the term under the square root, \( b^2 - 4ac \), known as the discriminant, tells us about the nature of the roots. If it's positive, there are two real and distinct solutions; if it's zero, there is exactly one real solution; and if it's negative, we have complex solutions.

The quadratic formula is a surefire way to get the solutions, especially when dealing with more complex quadratics, ensuring that students can confidently tackle any quadratic equation presented to them.

Factoring Quadratics

Factoring quadratics is akin to breaking down a complex structure into its basic building blocks. When you have a quadratic equation in the form of \( ax^2 + bx + c = 0 \), factoring involves finding two binomials that, when multiplied together, give you back the original quadratic equation. Factoring is a crucial skill because it is a straightforward method to solve for \( x \) when the quadratic can be easily decomposed into factors.

For example, if the given exercise \( 4x^2 - 3 = 57 \) had been in the format \( 4x^2 + 4x - 60 = 0 \), we might look for factors of \( -240 \) (which is \( 4 \times -60 \))) that add up to \( 4 \) (the middle coefficient). Once the quadratic is factored into \( (4x + \dots)(x + \dots) = 0 \), we can solve for the roots. It’s essential to remember that not all quadratics are factorable, but when they are, factoring offers a quick and elegant solution.

Graphing Quadratic Equations

Graphing quadratic equations provides a visual representation of the function and helps identify key features such as the vertex, axis of symmetry, and the roots or x-intercepts. A quadratic equation graph is a parabola, which can open upwards or downwards depending on the sign of the coefficient \( a \) in the standard form \( ax^2 + bx + c \).

The graph of a quadratic is symmetrical, and the vertex is the highest or lowest point on the graph depending on the direction of the opening. When given an equation, such as in our exercise \( 4x^2 - 3 = 57 \), we can complete the square to convert it into vertex form, or we can calculate the vertex using the formula \( (-b/2a, f(-b/2a)) \), where \( f(x) \) is the original quadratic equation. Knowing how to graph quadratics is incredibly beneficial, as it allows students to understand the behavior of the function and make predictions about the values of \( x \) for different values of \( y \) beyond just solving for the roots.