Solve common-factor and square-root quadratic equations

Which set contains every real solution?

The solid curve is y=x² and the dashed horizontal line is y=c. Move c from negative values through 0 to positive values. A crossing gives an input where x²=c.

What happens to the number of distinct real solutions?

The null factor law

For real numbers, a product is zero exactly when at least one factor is zero. The factors need not both be zero at the same input

1. 1Start with zero on one side The law does not apply directly to a product equal to a nonzero number
2. 2Keep every branch Do not divide by an expression that might be zero. Record its zero as a possible solution instead

Can the null factor law be applied directly?

After taking out x, what number fills the blank?

Which is the complete solution set?

Rearrange before factorising

Subtract the repeated constant from both sides. Then factor out x and solve the two branches

1. 1Solve both factors The constant factor 2 is never zero. The variable factors give x=0 or x=−3.

What is the smaller solution?

How many distinct real solutions are there?

Isolate the square

Divide by the nonzero coefficient before taking square roots. A positive squared value has both a negative and a positive root

1. 1Principal square root versus solving The symbol √16 means 4. The equation x²=16 asks for every input whose square is 16, so it has two solutions

After dividing by −3, what does x² equal?

Which set contains every real solution?

How many real solutions does this equation have?

The graph intersections for x²=9 have which x-coordinates?

Non-square values and numerical roots

An exact root need not be an integer. Keep both branches. Decimal approximations can be found by comparing squares or using a calculator

1. 1Bracket the positive root The positive square root lies between 2.23 and 2.24 \(\displaystyle \begin{gathered}2.23^2=4.9729\\2.24^2=5.0176\end{gathered}\)
2. 2Round only after enough precision The midpoint 2.235 has square 4.995225, below 5. Therefore the positive root is above the midpoint and rounds to 2.24 to two decimal places. The negative root rounds to −2.24

Give both roots to two decimal places

Which is the exact solution set?

Check in the original equation

Substitution checks a candidate. To establish all roots, also use a complete solving method with every branch retained

1. 1Equations and physical restrictions An unrestricted equation may have two roots. In a length problem a negative root can be rejected because a length must be positive. State the restriction rather than silently dropping a root

A square has area 49 cm². Which side length is physically possible?