The line \(y=kx-3\) meets the curve \(y=x^2+2x\) at two distinct points.

Find the set of possible values of \(k\), giving your answer in exact form.

Find the values of \(k\) for which the equation \(x^{2}+4 k x+k+3=0\) has two equal roots.

In this question, \(k\) is a constant. It is given that \(2 x^{2}+(3 k-2) x+k=0\) has roots that are real and distinct. Find the set of possible values of \(k\).

A curve has equation \(y=x^{2}-8 x+c\), where \(c\) is a constant. (a) Find the value of \(c\) in each of the following cases. (i) The curve crosses the \(x\)-axis at \(x=2\).

(ii) The minimum value of \(y\) is 3 .

(b) Find the range of values of \(c\) for which \(y\) is always greater than 0 .

Use algebra to show that the equation \(5 x(x-3)=5 x-26\) has no real solutions.

Find the values of \(k\) for which the curve

\(y=x^2+kx+4k-15\)

is completely above the \(x\)-axis.

Given that the equation \(k x^{2}+(2 k-1) x+k+1=0\) has no real roots, find the set of possible values of \(k\).

Do not use a calculator in this question.

Solve the equation

\((2+\sqrt5)x^2=4x+3(2-\sqrt5),\)

giving your answers in the form \(a+b\sqrt5\), where \(a\) and \(b\) are integers.

Find the values of \(k\) such that the line \(y=9kx+1\) does not meet the curve \(y=kx^2+3x(2k+1)+4\).

Find the possible values of \(k\) for which the equation \(kx^2+(k+5)x-4=0\) has real roots.

(a) Find the set of values of \(x\) for which \((5x-1)(6-x)\lt 0\).

(b) Show that, when \(k\ne-\frac12\), the equation \((2k+1)x^2-4kx+2k-1=0\) has distinct real roots for all values of \(k\).

Find the values of the constant \(k\) for which the equation

\(kx^2-3(k+1)x+25=0\)

has equal roots.

Find the possible values of the constant \(k\) such that the equation

\(kx^2+4kx+3k+1=0\)

has two different real roots.

Find the exact values of the constant \(k\) for which the line

\(y=2x+1\)

is a tangent to the curve

\(y=4x^2+kx+k-2.\)

Find the values of the constant \(k\) for which

\((2k-1)x^2+6x+k+1=0\)

has real roots.

Find the set of values of \(m\) for which the line \(y=mx-2\) does not touch or cut the curve

\(y=(m+1)x^2+8x+1.\)

Find the values of \(k\) for which the line

\(y=kx+3\)

is a tangent to the curve

\(y=2x^2+4x+k-1.\)

Do not use a calculator in this question.

Find the positive solution of the equation

\((5+4\sqrt7)x^2+(4-2\sqrt7)x-1=0,\)

giving your answer in the form \(a+b\sqrt7\), where \(a\) and \(b\) are fractions in their simplest form.

Find the values of \(k\) for which the line

\(y=kx-7\)

and the curve

\(y=3x^2+8x+5\)

do not intersect.

Find the set of values of \(k\) for which

\(4x^2-4kx+2k+3=0\)

has no real roots.

The curve \(y=2x^2+k+4\), where \(k\) is a constant, intersects the straight line \(y=(k+4)x\) at two distinct points. Find the possible values of \(k\).

Find the set of values of \(k\) for which the equation

\(x^2+(k+9)x+9=0\)

has two distinct real roots.

Do not use a calculator in this question.

Solve the equation

\((\sqrt7-2)x^2-4x+(\sqrt7+2)=0,\)

giving each answer in the form \(a+b\sqrt7\), where \(a\) and \(b\) are rational numbers.

Show that the line \(y=mx+4\) will touch or intersect the curve \(y=x^2+3x+m\) for all values of \(m\).

Find the set of values of \(k\) for which the equation \((k-1)x^2+kx-k=0\) has real and distinct roots.

Find the values of \(k\) for which the line \(y=kx-3\) and the curve \(y=2x^2+3x+k\) do not intersect.

Determine the set of values of \(k\) for which the equation \((3-2k)x^2+(2k-3)x+1=0\) has no real roots.

Show that the roots of

\(px^2+(p-q)x-q=0\)

are real for all real values of \(p\) and \(q\).

Find the set of values of \(k\) for which the equation \(kx^2+3x-4+k=0\) has no real roots.