Recognise, standardise and test quadratic equations

After simplifying this equation, is it quadratic?

Move the slider from 0 to 2. Filled cards represent positive terms; hollow cards represent subtracted terms. Compare the written x² terms with their combined coefficient. Card sizes do not represent areas

Moving through every position of the slider, when was the result linear?

Degree two after simplification

A quadratic equation is a polynomial equation whose highest remaining power is exactly 2. Collect all terms on one side first

1. 1Nonzero leading coefficient The squared-term coefficient can be positive or negative, but cannot be zero \(\displaystyle ax^2+bx+c=0,\quad a\ne0\)
2. 2Missing terms are allowed Either the linear or constant coefficient may be zero \(\displaystyle x^2-16=0\)
3. 3A squared term is not enough A nonzero cubic or higher-power term makes the degree greater than 2 \(\displaystyle x^3+x^2-4=0\)

Is this a quadratic equation?

What is the degree of this equation?

Standard form and signed coefficients

Use left side minus right side to collect the displayed equation into zero form. Keep each sign with its coefficient

1. 1Subtract the right side Subtract the same expression from both sides \(\displaystyle 2x^2+3=5x\)
2. 2Collect by descending powers The linear coefficient is negative after subtracting 5x \(\displaystyle 2x^2-5x+3=0\)
3. 3Read missing coefficients as zero No written linear term means a zero linear coefficient \(\displaystyle x^2-16=0\Rightarrow b=0\)

For the zero form shown, what is b?

Collect the equation as left side minus right side. What is c?

What is the leading coefficient a?

Which equation is the zero form obtained by subtracting the right side?

Evaluate before deciding whether a value is a solution

Substitution gives a value of an expression. A solution must make the original equation true

1. 1Bracket negative inputs Square the complete negative input before multiplying by its coefficient \(\displaystyle (-2)^2=4\)
2. 2Multiply each coefficient A coefficient multiplies; it is not added to the substituted value \(\displaystyle 2(-2)^2=8\)
3. 3Keep the linear sign A negative coefficient multiplied by a negative input gives a positive term \(\displaystyle -3(-2)=6\)

Evaluate the expression at x=−2

The expression equals 18 at x=−2. Is x=−2 a solution of this equation?

Does x=1 satisfy the original equation?

Does x=2 satisfy the same original equation?

Equivalent equations and coefficient conventions

Multiplying every term of an equation by the same nonzero number preserves its solutions. Coefficients belong to the particular form you are asked to use

1. 1Both equations have the same solutions The second equation is twice the first \(\displaystyle \begin{gathered}x^2-5x+6=0\\2x^2-10x+12=0\end{gathered}\)
2. 2Identify the displayed coefficients For the second equation, a=2, b=−10 and c=12. Do not silently divide when asked to read those coefficients
3. 3Recognition does not find every solution Testing one value checks that value only. The next sections develop solving methods and connections to graphs

Which description is correct after simplifying?

Which description is correct after simplifying?