Problem 29 - Algebra 1

Question

Solve the equation algebraically. Check the solutions graphically. $$ x^{2}-4=12 $$

Step-by-Step Solution

Answer

The solutions to the equation \(x^{2}-4=12\) are \(x = 4\) and \(x = -4\).

Step 1: Simplify the Equation

To begin solving this quadratic equation, it should first be rewritten in an easier to solve format. Subtract the constant 4 from 12 to get 8 and write it on the RHS: \[x^{2} - 4 - 12 = 0 \rightarrow x^{2} - 16 = 0 \]

Step 2: Finding the Roots

Applying the square root property, which is whenever you have a variable squared equal to a number, your solutions are going to be plus or minus the square root of that number. Hence, the roots of the equation are \[x=± \sqrt{16}\] Thus, the solutions are \(x = 4\) and \(x = -4\).

Step 3: Graphical Confirmation

To confirm graphically, plot the function \(f(x)= x^{2}-16\) against 'x' values. The graph is a parabola and it intersects the x-axis at points \(x = 4\) and \(x = -4\), verifying that these are indeed the solutions to the equation.

Key Concepts

Square Root Property

The square root property is a helpful tool in solving quadratic equations, specifically when dealing with simple forms like \(x^2 = a\). When a variable squared is set equal to a constant, you can solve it by taking the square root of both sides.
This means you'll get two solutions: one positive and one negative. For example, in the equation \(x^2 - 16 = 0\), you can isolate the \(x^2\) term to get \(x^2 = 16\).
By applying the square root property, you take the square root of 16, resulting in \(x = \pm 4\). This simply means \(x\) can be 4 or -4.

• Use this method when the equation is in the form \(x^2 = a\).
• Remember that each solution is valid, as both the positive and negative values satisfy \(x^2 = a\).

It's crucial to check both solutions unless other constraints are given in the problem.

Graphical Solution

Solving equations graphically provides a visual perspective on understanding solutions. For quadratic equations like \(x^2 - 16 = 0\), graphing helps confirm the solutions you find algebraically.
The graph of a quadratic function is a curve called a parabola. To sketch this, you write the function as \(f(x) = x^2 - 16\) and plot it on the coordinate plane.
This particular parabola opens upwards since it is in the form of \(ax^2 + bx + c\) where \(a\), the coefficient of \(x^2\), is positive.

• To find intersections, set the function to 0, meaning you look for where the graph intersects the x-axis.
• The x-intercepts of this graph are your solutions. These points confirm the algebraic answers of \(x = 4\) and \(x = -4\).

By visualizing, you reinforce the algebraic solutions and gain a better understanding of where solutions lie in the context of the problem.

Parabola

A parabola is the U-shaped graph that you get when you plot a quadratic function like \(y = ax^2 + bx + c\). In the example equation \(f(x)= x^2 - 16\), which simplifies to \(y = x^2 - 16\), the absence of \(bx\) shifts the vertex right on the y-axis.
This form is a standard parabola that opens upwards, indicating that the coefficient \(a = 1\), is positive.
The vertex of this parabola, or the lowest point on the graph since it opens upwards, is at (0, -16).

• The axis of symmetry, a key feature, is a vertical line that passes through the vertex, here it's \(x = 0\).
• Its utility lies in illustrating balanced solutions: the parabola is symmetrical about this axis, which helps verify solutions.

Understanding the shape and characteristics of a parabola aids you significantly in validating solutions for quadratic equations and exploring other properties of quadratic functions.