Problem 8 - Basic Technical Mathematics with Calculus

Question

Determine whether or not the given equations are quadratic. If the resulting form is quadratic, identify \(a, b,\) and \(c,\) with \(a>0 .\) Otherwise, explain why the resulting form is not quadratic. $$(T-7)^{2}=(2 T+3)^{2}$$

Step-by-Step Solution

Answer

Yes, it is quadratic with coefficients \(a = 3\), \(b = 26\), \(c = -40\).

Step 1: Expand the Left Side of the Equation

Expand \((T-7)^2\) using the formula \((a-b)^2 = a^2 - 2ab + b^2\). This gives:\[ T^2 - 2\cdot T\cdot 7 + 7^2 = T^2 - 14T + 49 \]

Step 2: Expand the Right Side of the Equation

Expand \((2T+3)^2\) using the formula \((a+b)^2 = a^2 + 2ab + b^2\). This gives:\[ (2T)^2 + 2\cdot 2T \cdot 3 + 3^2 = 4T^2 + 12T + 9 \]

Step 3: Set the Expanded Forms to an Equation

Write the expanded forms as an equation:\[ T^2 - 14T + 49 = 4T^2 + 12T + 9 \]

Step 4: Move All Terms to One Side of the Equation

Subtract \((4T^2 + 12T + 9)\) from both sides:\[ T^2 - 14T + 49 - 4T^2 - 12T - 9 = 0 \]

Step 5: Combine Like Terms

Simplify the equation by combining like terms:\[ -3T^2 - 26T + 40 = 0 \]

Step 6: Identify Quadratic Form

Recognize that the equation \(-3T^2 - 26T + 40 = 0\) is in standard quadratic form \(aT^2 + bT + c = 0\). Therefore, since \(a = -3\) is less than zero, this does not satisfy \(a > 0\). To ensure \(a > 0\), multiply the equation by \(-1\): \[ 3T^2 + 26T - 40 = 0 \]

Step 7: Extract Quadratic Coefficients

Identify the coefficients: \(a = 3\), \(b = 26\), \(c = -40\), with \(a > 0\).

Key Concepts

Standard Quadratic Form

Understanding the standard quadratic form is essential when dealing with quadratic equations. This form looks like this: \[ ax^2 + bx + c = 0 \] Here, "\(a\)", "\(b\)", and "\(c\)" are known as coefficients, and "\(x\)" represents the variable of the equation.
What makes this structure unique is the inclusion of the specific term, "\(ax^2\)", which is the square of the variable. This is what categorizes an equation as quadratic rather than linear or another form.Whenever working with quadratic equations, the first goal is often to rearrange or manipulate given expressions into this standard form. This makes it easier to solve and understand the properties of the equation.

Coefficients in Quadratic Equations

Coefficients are the numbers that multiply the variable and its powers in an equation. In a standard quadratic equation like \[ ax^2 + bx + c = 0, \] "\(a\)", "\(b\)", and "\(c\)" are the coefficients.

• "\(a\)" is the coefficient of the quadratic term \(x^2\). It must be non-zero. If "\(a\)" were zero, the equation would no longer be quadratic.
• "\(b\)" is the coefficient of the linear term \(x\).
• "\(c\)" is the constant term without any \(x\).

The sign and magnitude of these coefficients can significantly control the properties of the quadratic equation.For example, when rearranging an equation to ensure that the leading coefficient "\(a\)" is positive, it may be necessary to multiply through by \(-1\). Understanding how to manipulate these coefficients also allows us to determine the roots and the shape of the graph of the quadratic equation.

Expanding Algebraic Expressions

Expanding algebraic expressions is a critical skill in solving quadratic equations. This means taking expressions like \((T-7)^2\) or \((2T+3)^2\), and using known formulas to rewrite them as a sum of terms.
The formula \((a-b)^2 = a^2 - 2ab + b^2\) applies when the expression involves a subtraction. Similarly,\((a+b)^2 = a^2 + 2ab + b^2\) is used when the expression involves addition.Articles:

• In the example \((T-7)^2\), the expansion becomes \(T^2 - 14T + 49\).
• For \((2T+3)^2\), applying the expansion results in \(4T^2 + 12T + 9\).

Expanding expressions like these allows us to transform complex equations into a simpler, standard quadratic form, so we can identify coefficients and solve them efficiently. This technique can be practiced by repeatedly applying these standard identities until they become intuitive.