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Worked solution

Use two distinct real roots to restrict a parameter

The numbers xx and yy satisfy the equation

4x2+16xy+y2+16x+14y+13=04x^2+16xy+y^2+16x+14y+13=0.

If a real value of xx is substituted, the equation becomes a quadratic equation in yy. Given that two distinct real values of yy may be found from this equation, show that

5x2+8x+3>05x^2+8x+3>0,

and hence find the set of possible values of xx.

[2004/I/5]

CJC · Review Problems · Question 15

Answer

x<−1x<-1 or x>−35x>-\frac35.

Discriminant

Regard the equation as y2+(16x+14)y+(4x2+16x+13)=0y^2+(16x+14)y+(4x^2+16x+13)=0. Two distinct real roots require D=(16x+14)2−4(4x2+16x+13)>0D=(16x+14)^2-4(4x^2+16x+13)>0. Dividing by 4 gives 64x2+112x+49−(4x2+16x+13)>064x^2+112x+49-(4x^2+16x+13)>0, or 12(5x2+8x+3)>012(5x^2+8x+3)>0. Hence 5x2+8x+3>05x^2+8x+3>0, as required. Equivalently the undivided discriminant is 48(5x2+8x+3)48(5x^2+8x+3).

How did this part go?

Possible inputs

Factor: (5x+3)(x+1)>0(5x+3)(x+1)>0. The sign sequence across −1,−35-1,-\frac35 is +,−,++,-,+. Therefore {x∈R:x<−1 or x>−35}\{x\in\mathbb R:x<-1\text{ or }x>-\frac35\}. Both boundary inputs give a repeated root in yy, so both are excluded.

How did this part go?

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The discriminant factor (5x+3)(x+1) has signs positive, negative, positive across −1 and −3/5. Both zeros are open because two distinct real roots require strict positivity.
The discriminant factor (5x+3)(x+1) has signs positive, negative, positive across −1 and −3/5. Both zeros are open because two distinct real roots require strict positivity.
The discriminant factor (5x+3)(x+1) has signs positive, negative, positive across −1 and −3/5. Both zeros are open because two distinct real roots require strict positivity.

Common mistakes

Using D≥0 would admit repeated roots and incorrectly include both boundary values.

Solution notes

The diagrams are independently reconstructed mathematical equivalents. The plotted geometry is illustrative; use the labelled exact or approximate values and the stated domain.

No marks are printed for Q15.

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