Factorising and Solving Quadratics Flashcards (45 cards)

What does factorising do to an expression?

It writes the expression as a product of factors by taking out common factors.

When is a factorisation complete?

When all common factors have been taken outside the brackets.

What is the sequence for factorising by grouping?

Pair terms with common factors, factor each pair, then take out the common bracket.

How do you factor a monic quadratic $x^2+bx+c$ using two numbers?

Find numbers with product $c$ and sum $b$, then use them as the factor constants.

For $ax^2+bx+c$ with $a\ne1$, what product and sum guide the split-middle-term method?

The two numbers have product $ac$ and sum $b$.

After splitting the middle term in $ax^2+bx+c$, what does the grouping method factor out?

The common bracket.

What is the factorization rule for a difference of two perfect squares?

$A^2-B^2=(A-B)(A+B)$

How can a cubic $ax^3+bx^2+cx$ be factorised?

Take out the common factor of $x$, then factorise the remaining quadratic.

What form defines a quadratic equation, and what restriction applies to its leading coefficient?

$ax^2+bx+c=0$, where $a\ne0$.

In a quadratic equation, what is the greatest permitted power of the variable?

Two.

What values may the coefficients $b$ and $c$ take in a quadratic equation?

Either coefficient may be zero.

How many solutions can a quadratic equation have?

Zero, one, or two solutions.

What must be true when a product of factors equals zero?

At least one factor must equal zero.

What is the first step in solving a factorisable quadratic equation?

Rearrange it to equal zero.

After factoring a quadratic equation, what do you do before solving the resulting equations?

Set each factor equal to zero.

What formula solves $ax^2+bx+c=0$?

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

How do you rewrite $x^2+bx$ when completing the square for a monic quadratic?

$\left(x+\frac b2\right)^2-\left(\frac b2\right)^2$

After completing the square in a quadratic equation, what steps solve for $x$?

Isolate the square, then take both square roots.

What is the minimum value of $(x-h)^2+k$, and where does it occur?

The minimum is $k$ when $x=h$.

What sequence solves a quadratic word problem while checking contextual validity?

Define a variable, express relationships algebraically, form and solve the quadratic, then reject contextually invalid roots.

When must a mathematically valid root be rejected in a geometric or physical problem?

When it gives an impossible contextual quantity, such as a length.

When should substitution replace elimination in a linear–nonlinear simultaneous pair?

When one equation is linear and the other is nonlinear.

After substituting the linear equation into the nonlinear equation, what should you solve?

The resulting quadratic.

After solving the quadratic, how do you find each paired y-value?

Substitute each x-value back into an equation.

How does $ah+ak+bh+bk$ factorise by grouping?

$(h+k)(a+b)$

How does $x^2+6x+8$ factorise?

$(x+2)(x+4)$

When factoring a quadratic with leading coefficient 2 by inspection, what first terms do the factors use?

$2x$ and $x$

How does $4a^2-b^2$ factorise?

$(2a-b)(2a+b)$

What are the solutions of $x^2-25=0$?

$x=5$ and $x=-5$.

When should you use the quadratic formula?

When the equation cannot be solved by factorisation.

What does a square number under the quadratic formula’s radical indicate?

Factorisation was possible.

When rounding a quadratic-formula result, when should you round dependent values?

Only at the end, after calculating with exact x-values.

What are the exact solutions of $2x^2-3x-4=0$?

$x=\frac{3+\sqrt{41}}{4}$ and $x=\frac{3-\sqrt{41}}{4}$

Why does $x^2+6x+10=0$ have no real solutions?

Completing the square gives a square equal to a negative number.

What is the first step for completing the square when the $x^2$ coefficient is not 1?

Factor that coefficient from the quadratic and linear terms.

What are two positive numbers differing by 3 with a product of 88?

8 and 11.

How is a rectangle’s area calculated from its dimensions?

Multiply its length by its width.

When a quadratic formula is needed, how should x-values be used to find y-values?

Use the exact x-values.

When should coordinates be rounded in a simultaneous-equation problem?

Only at the end.

Which fields did Srinivasa Ramanujan contribute to?

Analysis and number theory.

In which year did Srinivasa Ramanujan travel to England to work with G. H. Hardy?

1914.

In which year did Srinivasa Ramanujan become a Fellow of the Royal Society?

1918.

What is Ingrid Daubechies best known for?

Her work on image compression technology.

What have Ingrid Daubechies’s image-processing methods helped scientists extract from bone and tooth samples?

Information.

What have Ingrid Daubechies’s methods helped establish about artworks?

Their age and authenticity.